Mathcad Professional 13.0 <description/> <author>Daniel N. Pope</author> <company/> <keywords/> <revisedBy>Daniel Pope </revisedBy> </userData> <identityInfo> <revision>13</revision> <documentID>da376cb8-7d48-40a9-9f39-435377af9c26</documentID> <versionID>c24aef64-09f8-4b9c-8459-223c50dfc491</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="dot" derivative="derivative" literal-subscript="small" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier New" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 5" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="current" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" imaginary-value="i" zero-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="72" right="72" top="72" bottom="72"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="237" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="110" left="6" top="11.25" width="108" height="15.75" align-x="23.25" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f size="14"> <b>Mathcad Tutorial</b> </f> </p> </text> </region> <region region-id="3" left="6" top="38.25" width="445.5" height="120" align-x="14.25" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This document presents a short tutorial for Mathcad. The instructions were originally written for use with Mathcad 13.<sp count="2"/>Mathcad 14 has some slight differences.<sp count="2"/>You should hopefully be able to translate the differences between Mathcad 13 and 14 since the terminology between the two programs is the same.<sp count="2"/>When in doubt, remember that the help files for any program are your best friend</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We will start with basic concepts related to the use of the software and then proceed to more advanced topics.<sp count="2"/>The discussions presented here will often request you to type a series of characters using the keyboard.<sp count="2"/>If I want you to type a series of characters, I will enclose them in quotation marks (" ").<sp count="2"/>If part of the typing involves special keys, like <b>Shift</b>, or <b>Space</b> <inlineAttr font-weight="normal">, the keys will be indicated in </inlineAttr> <b>Bold</b> <inlineAttr font-weight="normal"> letters with the first character capitalized.</inlineAttr> </p> </text> </region> <region region-id="112" left="6" top="176.25" width="96.75" height="12" align-x="16.5" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Order of Operations:</u> </b> </p> </text> </region> <region region-id="114" left="36" top="194.25" width="423.75" height="72" align-x="53.25" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad processes instructions from left to right and top to bottom.<sp count="2"/>Notice that there are pages to the right of this page.<sp count="2"/>Any operations specified on the page to the right of this page will be processed prior to the operations on the page below this one.<sp count="2"/>But it is not just page-by-page, but line-by-line that the processing takes place.<sp count="2"/>Thus an operation directly to the right of a line on the current page is processed prior to an operation located below the current line.<sp count="2"/>The fictional operations listed below would be processed in the order indicated based on their positions.</p> </text> </region> <region region-id="116" left="72" top="284.25" width="59.25" height="12" align-x="73.5" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1) operation 1</p> </text> </region> <region region-id="119" left="360" top="284.25" width="59.25" height="12" align-x="361.5" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">2) operation 2</p> </text> </region> <region region-id="121" left="216" top="314.25" width="59.25" height="12" align-x="217.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3) operation 3</p> </text> </region> <region region-id="123" left="324" top="314.25" width="59.25" height="12" align-x="325.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">4) operation 4</p> </text> </region> <region region-id="125" left="36" top="332.25" width="59.25" height="12" align-x="37.5" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">5) operation 5</p> </text> </region> <region region-id="127" left="144" top="344.25" width="59.25" height="12" align-x="145.5" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">6) operation 6</p> </text> </region> <region region-id="137" left="6" top="368.25" width="71.25" height="12" align-x="14.25" align-y="378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Text and Math:</u> </b> </p> </text> </region> <region region-id="143" left="36" top="386.25" width="420.75" height="120" align-x="43.5" align-y="396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The characters you type are placed wherever the cursor (the red plus <c val="#f00">+</c>) is located on the worksheet as you begin to type.<sp count="2"/>If you start typing and hit the <b>Space</b> bar before entering a character that performs a mathematical operation (*, =, :, etc.), then Mathcad assumes that you are entering text and creates a region to hold that text.<sp count="2"/>For example, this current text region was created by type "The" followed by <b>Space</b> and then continuing with the rest of the text.<sp count="2"/>Note that Mathcad has created a text region that is resizeable to hold this text.<sp count="2"/>If you click on this text, you will see the resizeable box that surrounds the text.<sp count="2"/>Once the box is visible, you can <b>Right-Click</b> and a pop-up menu appears.<sp count="2"/>One of the options is "Properties".<sp count="2"/>If you Left-Click on "Properties", you can highlight the region using a color of your choice (the default is yellow), or place a box around the region.<sp count="2"/> </p> </text> </region> <region region-id="145" left="36" top="524.25" width="411" height="36" align-x="45.75" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that the <b>Right-Click</b>/"Properties" options are also available for Math regions.<sp count="2"/>A math region is created any time you start typing and you hit a key that performs a mathematical operation (*, =, :, /, etc.) <b>before</b> you hit the <b>Space</b> bar.</p> </text> </region> <region region-id="184" left="36" top="578.25" width="425.25" height="24" align-x="45.75" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that you can easily tell the difference between a text and math region since they use different fonts.</p> </text> </region> <region region-id="181" left="0" top="618" width="6000" height="6" align-x="0" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="129" left="6" top="632.25" width="153.75" height="12" align-x="16.5" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Units and Pre-defined Variables:</u> </b> </p> </text> </region> <region region-id="131" left="36" top="656.25" width="418.5" height="96" align-x="53.25" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad allows you to associate units with any number that you enter.<sp count="2"/>These units are designated by letters or groups of letters, for example m is meters, J is joules, etc.<sp count="2"/>Thus, Mathcad has already assigned a certain meaning (a unit) to certain letters and groups of letter.<sp count="2"/>In general, you should not use letters as variables in your solution procedure that are already used by Mathcad as a unit.<sp count="2"/>I will illustrate this with a short example.<sp count="2"/>Let us say that I want to assign a value of "5" to a variable named "m".<sp count="2"/>To do this we type "m:5" (We will discuss how to assign values to variables in more detail when we address <b>Equal Signs</b>).<sp count="2"/>The following is displayed: </p> </text> </region> <region region-id="146" left="36" top="765" width="24.75" height="12.75" align-x="48.75" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="1"/> </region> <region region-id="148" left="36" top="794.25" width="423.75" height="72" align-x="45.75" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note the green squiggley line underneath the "m".<sp count="2"/>If you <b>Left-Click</b> on the green line, you will see the text "This expression redefines a built-in Mathcad unit."<sp count="2"/>Mathcad is warning you that for every calculation that follows this statement, the letter "m" means "5", not the unit meters.<sp count="2"/>Thus, if you wanted to specify meters later in your calculations, you could not use "m" because that means "5".<sp count="2"/>In fact, I will be using meters in future calculations in this tutorial, so I will have to redefine it now.<sp count="3"/> </p> </text> </region> <region region-id="149" left="36" top="879" width="45" height="12.75" align-x="48.75" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>100</ml:real> <ml:id xml:space="preserve">cm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="151" left="36" top="902.25" width="414.75" height="36" align-x="45.75" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that if you Left-Click on the green line now, the text "This expression redefines a previously defined variable."<sp count="2"/>Again Mathcad is warning you that you had already defined "m" and are now giving it a different definition.<sp count="2"/>In this case, I really wanted to redefine it. </p> </text> </region> <region region-id="153" left="36" top="950.25" width="405" height="24" align-x="53.25" align-y="960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad has some predefined variables that you may need, the most common of which is the gravitational constant "g".<sp count="2"/>If you type "g=", you will see the following:</p> </text> </region> <region region-id="154" left="36" top="987" width="71.25" height="18.75" align-x="46.5" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">g</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>9.80665</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> <unitReference unit="second" power-numerator="-2"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="3"/> </region> <region region-id="161" left="216" top="986.25" width="55.5" height="33.75" align-x="226.5" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">g</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">ft</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">s</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>32.174048556430442</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="foot"/> <unitReference unit="second" power-numerator="-2"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="4"/> </region> <region region-id="156" left="36" top="1034.25" width="420.75" height="51" align-x="48" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad uses SI units for its internal calculations, so the value for "g" is as shown on the left.<sp count="2"/>The<sp count="2"/>right math region displays "g" in ft/s<sup>2</sup>.<sp count="2"/>If you <b>Left-Click</b> on the left region (the one in SI units), hit Tab, and then type "ft<b>/</b>s^2", you will recreate what is displayed in the right math region above.</p> </text> </region> <region region-id="236" left="36" top="1100.25" width="429" height="24" align-x="45.75" align-y="1110" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Note</b> that all of the units that are defined within Mathcad are available using the "Insert Unit" icon (it looks like a little measuring cup) at the top of the Mathcad window.<sp count="2"/> </p> </text> </region> <region region-id="182" left="0" top="1140" width="6000" height="6" align-x="0" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="163" left="36" top="1148.25" width="422.25" height="132" align-x="60.75" align-y="1158" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Temperature:</b> <sp count="2"/>Mathcad uses absolute temperature for its internal calculations, which for SI units is Kelvin.<sp count="2"/>You can work with degrees Celsius, however you can't multiply the temperature in Kelvin by a fixed number to convert to degrees Celsius, so this is not a "standard" unit conversion.<sp count="2"/>In fact, if you <b>Left-Click</b> on the "Insert Unit" icon, and then select "Temperature" under "Dimension" in the pop-up dialog, the text (function) is by degrees Celsius in the scrollable list box at the bottom of the dialog box.<sp count="2"/>Note also that there is a temperature difference in degrees Celsius (a 1 degree Celsius temperature difference is equal to a 1 Kelvin temperature difference, but 1 degree Celsius is 274.15 K).<sp count="2"/>You must be careful when using negative temperatures in degrees Celsius.<sp count="2"/>Since the degrees Celsius "unit" is a function, you need to make sure that the function is applied to the entire negative number.<sp count="2"/>I now show what happens if you do this incorrectly.<sp count="2"/>Let us say that the actual temperature is -5 deg C.</p> </text> </region> <region region-id="169" left="36" top="1304.25" width="36" height="12" align-x="49.5" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>correct:</b> </p> </text> </region> <region region-id="170" left="108" top="1304.25" width="51.75" height="15.75" align-x="123.75" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="1">T</ml:id> <ml:apply fixity="postfix"> <ml:id xml:space="preserve">°C</ml:id> <ml:parens> <ml:real>-5</ml:real> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="178" left="252" top="1305" width="60.75" height="15" align-x="267" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="1">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>268.15</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kelvin"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="6"/> </region> <region region-id="172" left="36" top="1352.25" width="45" height="12" align-x="53.25" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>incorrect:</b> </p> </text> </region> <region region-id="173" left="108" top="1352.25" width="51.75" height="15.75" align-x="123.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="1">T</ml:id> <ml:apply> <ml:neg/> <ml:apply fixity="postfix"> <ml:id xml:space="preserve">°C</ml:id> <ml:parens> <ml:real>5</ml:real> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="174" left="252" top="1353" width="66" height="15" align-x="267" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="1">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-278.15</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kelvin"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="8"/> </region> <region region-id="177" left="342" top="1352.25" width="57" height="12" align-x="342.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>impossible! </b> </p> </text> </region> <region region-id="180" left="36" top="1394.25" width="421.5" height="48" align-x="51" align-y="1404" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Caution:</u> </b> when using English units, be careful of pound-force (a unit of force) and pound-mass (a unit<sp count="2"/>of mass).<sp count="2"/>In Mathcad, <b>pound-force</b> is denoted by <b>lbf</b> and <b>pound-mass</b> is denoted by <b>lb</b>.<sp count="2"/>If you have any doubt as to how to specify a certain unit, go to the main menu and select "Insert", and then "Unit".<sp count="2"/>A list of dimensions and there associated units will appear.</p> </text> </region> <region region-id="6" left="6" top="1466.25" width="418.5" height="24" align-x="13.5" align-y="1476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Equal Signs:</u> </b> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <sp count="10"/>One of the first things you need to know is that there are different types of equal (=) signs:</p> </text> </region> <region region-id="8" left="36" top="1508.25" width="411.75" height="48" align-x="46.5" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When you use colon (:) it places the symbol := on the worksheet.<sp count="2"/>The := sign indicates that a value (with units) is being assigned to a variable.<sp count="2"/>For example if a pipe diameter is 0.5 m, we can assign this by typing "D:0.5m".<sp count="2"/>Note that the := symbol is also available on the Evaluation<sp count="2"/>Toolbar.</p> </text> </region> <region region-id="9" left="48" top="1569" width="38.25" height="12.75" align-x="60.75" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>0.5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="11" left="36" top="1592.25" width="400.5" height="96" align-x="60" align-y="1602" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Calculations are also done with the colon operator.<sp count="2"/>For example, if we want to calculate the area for flow through the pipe, we would type "A:<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">/4</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">Space</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">*D^2".<sp count="2"/>Note what happens when you hit the (space) bar.<sp count="2"/>As you type an equation, part or all of the equation will have a blue bar underneath it.<sp count="2"/>The portion of the equation with the blue bar underneath indicates what the next keystroke will operate on.<sp count="2"/>So, when you type </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, there should be a red bar under </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10">, indicating that your next keystroke (/) will operate on </f> </inlineAttr> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>After typing the 4, the red bar will only be under the 4.<sp count="2"/>If you DO NOT hit space, your next keystroke will operate on the 4, not on the quantity </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">/4.<sp count="2"/>Thus without hitting space you would get </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">/(4*D^2).</inlineAttr> </f> </p> </text> </region> <region region-id="23" left="36" top="1706.25" width="81" height="12" align-x="52.5" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Without Space bar</p> </text> </region> <region region-id="24" left="186" top="1706.25" width="67.5" height="12" align-x="195.75" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">With Space bar</p> </text> </region> <region region-id="20" left="48" top="1729.5" width="47.25" height="34.5" align-x="61.5" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="186" left="198" top="1729.5" width="40.5" height="28.5" align-x="211.5" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="185" left="0" top="1770" width="6000" height="6" align-x="0" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="33" left="36" top="1784.25" width="405" height="63" align-x="50.25" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If the value of "A" has been defined, typing "A=" will report the assigned value for A in the default units, i.e. it will evaluate "A".<sp count="2"/>If the value of "A" has not been defined, typing "A=" will produce "A :=" with a box for entering a value.<sp count="2"/>Since A is a defined area, the default units will be m<f family="Arial" charset="0"> <inlineAttr line-through="false"> <sup>2</sup> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">.<sp count="2"/>If you don't want m</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <sup>2</sup> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false"> and instead want ft</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <sup>2</sup> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">, just hit Tab to go to the end and type "ft^2".<sp count="2"/>Note that the equal sign (=) is also available on the Evaluation Toolbar. </inlineAttr> </f> </p> </text> </region> <region region-id="34" left="36" top="1863" width="51.75" height="18.75" align-x="48.75" align-y="1878" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">A</ml:id> <ml:unitOverride> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">ft</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>2.1134888680646196</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="foot" power-numerator="2"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="35" left="36" top="1898.25" width="54" height="33.75" align-x="48.75" align-y="1914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ρ</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1000</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kg</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">m</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="36" left="126" top="1898.25" width="42" height="27.75" align-x="138" align-y="1914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">V</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.5</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">m</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="37" left="36" top="1952.25" width="57" height="13.5" align-x="50.25" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m'</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ρ</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:apply> <ml:id xml:space="preserve">A</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="38" left="216" top="1946.25" width="78" height="27.75" align-x="229.5" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m'</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">lb</ml:id> <ml:id xml:space="preserve">hr</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>2337533.8535490688</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="hour" power-numerator="-1"/> <unitReference unit="pound"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="40" left="36" top="1994.25" width="404.25" height="63" align-x="43.5" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The third type of equal sign is obtained from the Boolean toolbar.<sp count="2"/>You can also get this equal sign by holding down the <b>Ctrl</b> key while you type "=".<sp count="2"/>This type of equal sign indicates a general equality and is used quite often in solve blocks.<sp count="2"/>For example, if you want to solve the quadratic equation x<f family="Arial" charset="0"> <inlineAttr line-through="false"> <sup>2</sup> </inlineAttr> </f>-4x+3=0, we could do the following (see Mathcad help for "solve blocks" to view examples):</p> </text> </region> <region region-id="42" left="36" top="2072.25" width="86.25" height="12" align-x="48" align-y="2082" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Guess a value for x:</p> </text> </region> <region region-id="43" left="72" top="2091" width="21.75" height="12.75" align-x="81.75" align-y="2100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="48" left="36" top="2109" width="24" height="12.75" align-x="47.25" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Given</ml:id> </math> <rendering item-idref="18"/> </region> <region region-id="50" left="126" top="2108.25" width="337.5" height="24" align-x="135.75" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type the word "Given" (Do Not type a space after Given or you will get a "text" Given and not a "math" given)</p> </text> </region> <region region-id="52" left="72" top="2145" width="68.25" height="18.75" align-x="129" align-y="2160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:real>3</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="54" left="180" top="2150.25" width="234" height="12" align-x="189.75" align-y="2160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type in the equation using the Ctrl= type of equal sign</p> </text> </region> <region region-id="57" left="36" top="2174.25" width="409.5" height="36" align-x="43.5" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Use a solve block function to solve the equation based on your initial guess.<sp count="2"/>Since we are solving a system of equations (one equation), we would usually use either the "Find" function or the "Minerr" function. </p> </text> </region> <region region-id="58" left="36" top="2223" width="57" height="12.75" align-x="56.25" align-y="2232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ans</ml:id> <ml:apply> <ml:Find auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="59" left="144" top="2223" width="36.75" height="12.75" align-x="163.5" align-y="2232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ans</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="62" left="36" top="2252.25" width="393" height="24" align-x="45.75" align-y="2262" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that ans = 3 if we use an initial guess of x:=5 above, and ans=1 if we use an initial guess of x:=0.5 above.</p> </text> </region> <region region-id="187" left="0" top="2286" width="6000" height="6" align-x="0" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="71" left="6" top="2294.25" width="51.75" height="12" align-x="27" align-y="2304" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Subscripts:</u> </b> </p> </text> </region> <region region-id="73" left="36" top="2318.25" width="387.75" height="24" align-x="41.25" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To add a subscript to a variable name, use the "period" key (.).<sp count="2"/>For example, type "p.0:10psi" to get</p> </text> </region> <region region-id="76" left="54" top="2355" width="43.5" height="15" align-x="69" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="0">p</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="78" left="36" top="2384.25" width="404.25" height="75" align-x="41.25" align-y="2394" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To add subscripts to variables within a text region, use the Formatting Toolbar.<sp count="2"/>Select "View" and then "Toolbars" from the main menu and make sure that "Formatting" is checked.<sp count="2"/>The formatting toolbar should be located in the region between the main menu and the worksheet.<sp count="2"/>To create a subscript within the text region, select the <b>x</b> <c val="#00f"> <sub>2</sub> </c> symbol on the Formatting toolbar.<sp count="2"/>You can either highlight text and select the subscript symbol, or select the subscript symbol, type the text and deselect the subscript symbol to return to normal text:</p> </text> </region> <region region-id="80" left="36" top="2480.25" width="423.75" height="87" align-x="39" align-y="2490" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A third type of subscript is associated with vectors and matrices.<sp count="2"/>This subscript is accessed from the Vector and Matrix Toolbar through the option X<sub>n</sub> (open the Vector and Matrix Toolbar and find this option).<sp count="2"/>You can also create vectors and matrices using the Vector and Matrix Toolbar.<sp count="2"/>For example, type "y:", open the Vector and Matrix Toolbar and select the top-left option to create a Matrix or Vector.<sp count="2"/>A dialog box appears that asks how many rows and columns you want.<sp count="2"/>Enter 3 rows and 1 column and hit "OK".<sp count="2"/>An empty column vector with three elements should appear.<sp count="2"/>Place the values 3, 4, and 5 in the vector as shown below:</p> </text> </region> <region region-id="81" left="54" top="2578.5" width="37.5" height="45.75" align-x="65.25" align-y="2604" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">y</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>3</ml:real> <ml:real>4</ml:real> <ml:real>5</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="83" left="36" top="2636.25" width="420.75" height="66" align-x="53.25" align-y="2646" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad starts the Matrix and Vector numbering with the number ZERO (0).<sp count="2"/>Therefore, the first element in our matrix is y<sub>0</sub>.<sp count="2"/>The Matrix toolbar must be used to access values within a vector/matrix.<sp count="2"/>For example, if we wish to access the second element of our vector, which has a value of 4, we would type "y", then select X<sub>n</sub> on the Matrix toolbar, type "1" for the subscript, then type "=" to get</p> </text> </region> <region region-id="84" left="54" top="2721" width="32.25" height="18" align-x="69" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="196" left="0" top="2754" width="6000" height="6" align-x="0" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="198" left="6" top="2762.25" width="102" height="12" align-x="21" align-y="2772" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Vectors and Matrices:</u> </b> </p> </text> </region> <region region-id="202" left="36" top="2786.25" width="423" height="168" align-x="43.5" align-y="2796" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can create and manipulate vectors and matrices using the Vector and Matrix Toolbar.<sp count="2"/>To access the Vector and Matrix Toolbar click on the icon at the top of the Mathcad window that looks like 9 dots between two square brackets [] (click on it now).<sp count="2"/>If you mouse over the 12 options within the toolbar, you will see a short description of each option.<sp count="2"/>I will discuss the options that I use most in this section.<sp count="2"/>The "Matrix or Vector" option is the top left option.<sp count="2"/>Note that the keyboard shortcut is also displayed when you mouse over "Matrix or Vector".<sp count="2"/>Holding down the <b>Ctrl</b> key and typing M is the same as clicking on the "Matrix or Vector" option. This option allows you to create a matrix or vector.<sp count="2"/>When you click on it you can create a matrix or vector by specifying the number of rows and columns in the dialog box that appears.<sp count="2"/>To create a column vector named "v" consisting of 3 elements, first type "v:", then selecting the "Matrix or Vector" option in the upper left corner of the Vector and Matrix Toolbar. In the dialog box that appears, enter 3 rows and 1 column and hit "OK".<sp count="2"/>The vector will appear with empty spots for each of the three elements.<sp count="2"/>I will enter 1, 2, and 3 for the values of the three elements.<sp count="2"/>The result is shown below.</p> </text> </region> <region region-id="203" left="36" top="2968.5" width="37.5" height="45.75" align-x="47.25" align-y="2994" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">v</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>1</ml:real> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="205" left="36" top="3032.25" width="419.25" height="60" align-x="43.5" align-y="3042" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The second option on the top row in the Vector and Matrix Toolbar allows you to access individual elements of a vector or matrix.<sp count="2"/>It is the "Subscript" option.<sp count="2"/>We have already discussed this option when reviewing the different types of subscripts.<sp count="2"/>Note the keyboard shortcut for the option, which is the left square bracket [, is shown when you mouse over the option. </p> </text> </region> <region region-id="207" left="36" top="3110.25" width="422.25" height="36" align-x="43.5" align-y="3120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The third option on the top row is the "Inverse" which gives the inverse of a matrix (if it exists).<sp count="2"/>You can also do this without the toolbar by raising a matrix to the "-1" power.<sp count="2"/>This option is useful when solving matrix equations.</p> </text> </region> <region region-id="209" left="36" top="3164.25" width="417" height="48" align-x="43.5" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The third option on the second row of the Vector and Matrix Toolbar takes the "Transpose" of a vector or matrix.<sp count="2"/>It looks just like raising the matrix to the "T" power, but it is not.<sp count="2"/>If I try to assign<sp count="2"/>the vector "v" (defined above) raised to the "T" power to a new variable "w" by typing "w:v^T", I would get the following<sp count="2"/> </p> </text> </region> <region region-id="210" left="36" top="3231" width="32.25" height="18.75" align-x="49.5" align-y="3246" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_dimensions Unitless Magnetic Flux Density " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="212" left="36" top="3266.25" width="423" height="48" align-x="45.75" align-y="3276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that the "T" is red, indicating an error.<sp count="2"/>If I <b>Left-Click</b> on <c val="#f00">T</c>, I get the message "This value has units: Magnetic Flux Density, but must have units: Unitless."<sp count="2"/>It turns out that "T" is the unit Tesla<sp count="2"/>in Mathcad (you might remember that unit from Physics 2).<sp count="2"/>If I do this the correct way, by typing "w:v" and selecting the "Matrix Transpose" option in the Vector and Matrix Toolbar, I get<sp count="4"/> </p> </text> </region> <region region-id="213" left="36" top="3327.75" width="31.5" height="18" align-x="49.5" align-y="3342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">w</ml:id> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">v</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="214" left="108" top="3333" width="62.25" height="12.75" align-x="120.75" align-y="3342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">w</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="1" cols="3"> <ml:real>1</ml:real> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="235" left="36" top="3362.25" width="87" height="12" align-x="46.5" align-y="3372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Other useful options</p> </text> </region> <region region-id="223" left="36" top="3386.25" width="66" height="12" align-x="48" align-y="3396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Range Variable</p> </text> </region> <region region-id="225" left="36" top="3410.25" width="51.75" height="12" align-x="42.75" align-y="3420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Dot Product</p> </text> </region> <region region-id="227" left="36" top="3440.25" width="62.25" height="12" align-x="47.25" align-y="3450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cross Product</p> </text> </region> <region region-id="229" left="36" top="3464.25" width="55.5" height="12" align-x="60" align-y="3474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Determinant </p> </text> </region> <region region-id="216" left="36" top="3488.25" width="138" height="12" align-x="44.25" align-y="3498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section under construction!</p> </text> </region> <region region-id="200" left="6" top="3524.25" width="84" height="12" align-x="24" align-y="3534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Graphing Results:</u> </b> </p> </text> </region> <region region-id="219" left="36" top="3554.25" width="60.75" height="12" align-x="47.25" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Graph Toolbar</p> </text> </region> <region region-id="221" left="36" top="3578.25" width="138" height="12" align-x="44.25" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section under construction!</p> </text> </region> <region region-id="231" left="6" top="3614.25" width="83.25" height="12" align-x="22.5" align-y="3624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Calculus Toolbar:</u> </b> </p> </text> </region> <region region-id="233" left="36" top="3644.25" width="138" height="12" align-x="44.25" align-y="3654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section under construction!</p> </text> </region> <region region-id="192" left="6" top="3674.25" width="51.75" height="12" align-x="26.25" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Symbolics:</u> </b> </p> </text> </region> <region region-id="193" left="36" top="3692.25" width="148.5" height="12" align-x="44.25" align-y="3702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section is under construction!</p> </text> </region> <region region-id="194" left="36" top="3725.25" width="68.25" height="48" align-x="76.5" align-y="3750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:integral/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:symResult> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="195" left="132" top="3726" width="76.5" height="40.5" align-x="180.75" align-y="3750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:integral/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:symResult> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="30"/> </region> <region region-id="96" left="6" top="3788.25" width="101.25" height="12" align-x="19.5" align-y="3798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Output Data to a File:</u> </b> </p> </text> </region> <region region-id="98" left="36" top="3812.25" width="416.25" height="48" align-x="54.75" align-y="3822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Although Mathcad is capable of creating graphs that are usable for visualization purposes, there may come a time when you wish to use another program to create more complex or "prettier" graphs.<sp count="2"/>An easy way to do this is to output data from Mathcad to a text file that can be read by another program.<sp count="2"/> </p> </text> </region> <region region-id="99" left="36" top="3872.25" width="149.25" height="12" align-x="36" align-y="3882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To export data once from a matrix:</p> </text> </region> <region region-id="102" left="54" top="3896.25" width="403.5" height="24" align-x="54" align-y="3906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">1.<sp count="2"/>Display the results as an . Large matrices display this way automatically. For small arrays, you may have to specify the display from the<sp count="2"/>dialog.</p> </text> </region> <region region-id="104" left="54" top="3932.25" width="402" height="24" align-x="54" align-y="3942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">2.<sp count="2"/>Right-click on the corner of the output table and choose Export from the resulting menu to bring up the Write to File dialog box.</p> </text> </region> <region region-id="106" left="6" top="3968.25" width="102.75" height="12" align-x="30.75" align-y="3978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Augmenting a Matrix:</u> </b> </p> </text> </region> <region region-id="108" left="36" top="3992.25" width="80.25" height="12" align-x="52.5" align-y="4002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">augment() function</p> </text> </region> </regions> <binaryContent> <item item-id="1">iVBORw0KGgoAAAANSUhEUgAAACEAAAARCAYAAABTnsXCAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AJZJREFUSEvllIEKwCAIRN2v78e3alPOiszRkDEhVjT04Z1tRwqKjgwRHRQNUJT4NUSyYfbi taI6kYtzhEBwB7oQTIc/IXHdtdHdTIelntCARvfbUfTi/Sip0rfKY8E0nlA6oWawt5J675dB fLoTbk/0TPjEmNi10BFtpkxodv1kkHH2GnA4XWUEU0EsOnN+BYJhPN9VICcko3Cel6ZNiQAA AABJRU5ErkJggg==</item> <item item-id="2">iVBORw0KGgoAAAANSUhEUgAAADwAAAARCAYAAACBzs+aAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ANdJREFUWEftlYEOgCAIRO3X+/FKl4UEie4mbebWmhbg4xCX7RhhphGBZxphJthUzT/woAwc beMRKa7lh37U1nu26qKwBsUhz4ZacEmJagF3AbaAZDAOCAWmQbSAkgotGc7/1kCswFK5v3Fc CnNDWna1rFJb63lDAFMfHFKDLkpaciCVX4+i3AYNrFUeZ4IAf0HhocA9iqMVtlRnEoY2EX5u pXOAaFqt9+1bX+BXXK0XuV1LPVWBsLkVXkv2UJkjgnv4SJQRjgJa5h6bRcS8gDN4yxuxgdE+ duXaTRYY+00yAAAAAElFTkSuQmCC</item> <item item-id="3">iVBORw0KGgoAAAANSUhEUgAAAF8AAAAZCAYAAABXTfKEAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AWJJREFUaEPtmIsOwjAIRef/f7RujTOs43HpwMZJExMzKW0Pt8B8PNex1JhDYINfYw6BZc6y tWrLOIVhHoGCH8h+LRxb/WwfZGBWiKc/t+mBIwEo+EmiKfhJYBG3BR+hlGRT8JPAWm4R8NVq WhSV36XOBgVf8C/A56ZS8EgQvtLtIP2vpqS+d6a2nr46mPXB3cie0uEj/a9k432eCTfD90/C pyCQ650BLsLnCf7I9dE2gii/FZ/3K7mWNznQEnzqj/PthYekTrdPTUXawdAgofBpAPY9IXOl gNBa0H93QyL/1UTetIPykcN6N75DtZQzonwNhHSDrJsidTGR0D/i6hezIHEw0Y4DPbiUJjzi 8MJHRIWeE/F16vMzoiulECTdWbAjlY/WrUhGp7TT53I0ipKdppZenZytdBMt8FrO9wJEssEI JzXnc0VwZJGawxMo+BOVYfb5E/d2+6XT33BvT/DCAV8q7IBR7IQG/wAAAABJRU5ErkJg gg==</item> <item item-id="4">iVBORw0KGgoAAAANSUhEUgAAAEoAAAAtCAYAAAAJIf2TAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AbJJREFUaEPtmgmOwjAMReH+h2ZtIU29vAItBf9KIyFqYvsl3qI5ni7PQU9O4ApKT07gkItI 4hZ1wsAICNQ9Rz/+PGzlQV0hjc8IzIIlUA2ooQMwD1VpUG3I9Z97WqVBWSeoDcUWlkAp9GDZ F6j3QPUhWDr0rP7JaxE2AeU1dKTR8xJuVqWiUs/O2lRqdVCzIzzkBO/7WVkeuuZJBQJ5JWoe dwnKctzsfDvnIzDZu3ETvFL/EVBWE/bKwmvDaMcObzNWA7U0HEieQHNUcJqyXBPZ/HVQ75ww y/jMIbdbTnKVpysaXbyImiVzUok8RQQgPbVZHqLVMNsEYvNNFzGILkbk+muNKNdk7zII2Xti 78OGHtSSvEMUeetFRaOHaZV6AoHIEB/QifqkMmrUHuXS0BOo+7aFybwiJK+YrT7C7DGMPJvC nuyXHNna1klR2Vr5L+kTKLhbAgVAzfIV+E05EXNGLEchcdgdsQTqSSAcqwSKEVDDyTjV/v8o cvdmXrNAuH8h5iVtd7z5C69fcMK644qWKZ+jKLCyoBR6MAyXJHLz4g7qKSdWNvSW7rRAQWIC BUGdAYlnfCgyz8kgAAAAAElFTkSuQmCC</item> <item item-id="5">iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVCAYAAAAQAyPeAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AQdJREFUWEftVosKxCAM2/3/R+9xTFGvNsnVOxhUGAxmY5umca/9XFuunoGLlFw9A1sS8slA kmKoIknxSDmd5jJc8/nXiF3ns2vMFcWV/eO+Fue+dPaaRZvQ7B0dHPmuEFKSb8/z4tnaKkFW IWqCETKsAhGe13VUT6ltVuMbG4GgBNluzXAiDbBi3WKb8XSVtZKUgqV4U4QUdYzYBkpKmclW UZNldJ75jUaI4lmSlyqFOfTJSvnKUxhSFOVE8BRPYWyimjC6t5Fc2TldZbTqOKEGWXj839KN HunsKmJ+joOYVJSiYkVVtjK+w1IKQdJVsJgZZ/FWq1ceHzbRJ+9LUozuJSlJCjfUB9JlKvAf hj9hAAAAAElFTkSuQmCC</item> <item item-id="6">iVBORw0KGgoAAAANSUhEUgAAAFEAAAAUCAYAAAD1GtHpAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ASpJREFUWEftl4EOgzAIRN3/f/SmZi5IgOOELIujiYmulJbXg3aP59qWaTUCG8RpNQJLbfiM 3jN5MNQJfCCuRWGrjeZTn8b2IOezLI7+ow/ZS7vI3+ZHNstvdq6TEqVj770TphWIDoz51qA9 iF4MaD1R7GY6a4ed8DJBWPMzQXrjo7gqwvk5iEcwOpXfV7F9D9AmZ2FZSke+TYWzsg9lfbGm ahVEqsgEiWwipaKxZYiWOjpSPQtNqhRtJtv/1XRGqSJPtQx0VOs8cGgdFYiZknEqBWw6X5E7 c5h4UBHs6CTP9GUOMzcO3SGVxAJmUjtSrLeG6Hd9xbHqrCcAy69eH33FYVOBgXdHW/pvX2c6 3wUoBRGl+l2gsHFQEFnn/2I/EBt2eiAOxAYCDS5e8VY2AX1BFs8AAAAASUVORK5CYII=</item> <item item-id="7">iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVCAYAAAAQAyPeAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AThJREFUWEftmIsOwiAMRfHX/fE5FiGl9nVZlxADidFJaS+HthhfxznKHiOBCmWPkUDZQH4J bChCVmwoFpSz09SGK75WKLGqjQ6u1dPY7Lkd9fO9dI4eiQbVPnuBn5rnQJp4DkmLH91bByQ5 kkQ8tWHPrwYkAqr5lqBoe7zAZELJhun5Q8FEKyAVCj0ZqT95WSHVu7dG6jXeGqn8hlJEMkVr VhERVr1rAL1MQXtLuAchUDzC/5ApUz0lcnp3Mid6mtoBIfq00vvpKfzeRuocERQBZ9081u0R 8S1ldf8OcRAtH9SnZY+ARmzNmJ3Om/1iVJ5b4MLmM0Eg2UlPPEvDRaJukG5Se+6lJdhnCbJu qZm5GV0dSoNz531GwIprSiRDaIlZ9itucEbTAIVmCdprZoKvumb/nyKczAc8sNSSHZabJwAA AABJRU5ErkJggg==</item> <item item-id="8">iVBORw0KGgoAAAANSUhEUgAAAFgAAAAUCAYAAAAJD/ojAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AS9JREFUWEftl9sSgjAMRPH/P1rBsUwpm8uWxAdIZxwRQi6n6ba+3utYauQR2ADXyCOw5Lku z191KAy5BHbAqwhtWgw/uSlg730uzQLlh55p+W4+0EDxfvvTzoSNdejgPrB0/S/QI4T2m70/ 5tsgSoCl+qS4Hh5wOqVZ9jjMsEH5aE2g5S/58r7DsrkF4LaU+29vN/Z2mnx4fMOVwS4XS980 nZzpbqtoDyDGRutwtnvFU4S1EcyAkvRwnBBkx9yzIMw8v7In0RJhaVVkB3u7l9mErgJmpSIU cERn90chdI0KjALsnVAG8gmwdB6Uio2E2seQpMOSLw02qs1T73i8s2TtoPksIGuJsf7ubk// VS7AXEtQgK3lxIV+hjUF+BlIYqsswLE8T94KcAFOJpDs/gPiWkUOvZEJPgAAAABJRU5ErkJg gg==</item> <item item-id="9">iVBORw0KGgoAAAANSUhEUgAAADMAAAARCAYAAABwxZQXAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ALlJREFUSEvdltEOgCAIRe3/P7rUhUNCEdHlYOslBTxcb+u6YwQvkWC8RPACkm+YS5hom+Sd z7MDFvfp1R/dBzUqZVIyDe6dBZDW69XX9hZh3q+d5fxV7igMqKJpbILRTo4bjFSjBY9hYY8J ptxVhd9GlcGKlMOiPjCYao1L2ukbCwxVFtfKSq2A0XwJf4WR7rTGoCs9w6lUlNFMF3tFC4Pv umRubp0av+mZmYOdlOPzd+akCc+exZUyD7V8jQ9mTU7BAAAAAElFTkSuQmCC</item> <item item-id="10">iVBORw0KGgoAAAANSUhEUgAAAD8AAAAuCAYAAACf1cHhAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AYRJREFUaEPtmYESgyAIht2r78W36rJTFANUTg6727VaJh8/IrrP7ziC1+OE93oEr+BXxG94 px7YyjsVfo95U8IfJclZl2QfKYCpMX9C30VZdnYBHyGhEza8wAOmwj6GfKp8/C5gt5ftU9iY +CTgzfK2x6NSY7TboWHvFn5UNtVWkttfVXkOvOUIKeAhDBWuVnlR23IVG/V8FX5U+dgyEnOW 1v0i22NKURQ0r3wP/KhQ1HzPE/apcqkB2H1YZ2saPaovc+XtKPBszIdv7gfu9UijtN51EZ+g KSz3WstYaT/okI7wPWepURrtWnVL6FEcDg0NmN4+8lWhs7Gewfd60lp7t/DF+LemnNTeWvXq osiBW19PdSr15Ix2lAUUVnpTynO457eM8tLNyGo4339uvAm0BDxn5wgC9axEN/xbeMz8HUtG 1D7NKk/dL2wlQtPwte0vqurXYgxJbpSZY4kx/8y7xCwNpztqEiye43h59rMUtbjzfMvmpZSf 7dylld/wih5wHfZ/TZQTjQvWYKQAAAAASUVORK5CYII=</item> <item item-id="11">iVBORw0KGgoAAAANSUhEUgAAADYAAAAmCAYAAACPk2hGAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AWlJREFUaEPtl40OgyAMhN2r78WdopBSy99RhBFMFuM2ar9re+JnP45txuMEm/HYZoQyXbjA /kyBVbHRC3YY++nu5jPNjFkYK74B7F0JqjRVvCav7mCubWz73OcaqHvD0b9iNhF6XmCCAsOY h+dkxNWQqlETGcI8PDcD58yDirkit1BEQY01kmuK9k6eY1FXHAUsZCqp/MRW5DasoXxNjBBE DK4aLKVcDRCdPSlOEZjUvznJ5cxCTpwSgGIwniSaUGxdSAhpW1XdikgAbtUtREHy8mYMCdCi mjymlFdqth0YVZoGDn2fGmwtYHR2u+88tAR4VNkp//UZt8LrVgmicQ3NCUFBSq/Rm+fMEhrb gVlA5IzeXJpljVjXNoy0XGmleLuiSbXYwnlgtFq5s4fC8GpdKut5mV4kkJC/R4FhHsu6gqH7 0hz47mDSAzgn8dR/uoI1t/sU/Vu/T2UerUQbqhU1IReYpppvxPoBcBZWkwtwS0sAAAAASUVO RK5CYII=</item> <item item-id="12">iVBORw0KGgoAAAANSUhEUgAAAEUAAAAZCAYAAABnweOlAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ATFJREFUWEftmeEOgyAMhNm7772ZLsJKudKi44fxlpgt6EA+rteKr5RS3g5+FIGc85uHYLDz IRAlCkIBUUIoG5TDV7/fu5U8Hoq2j5J46CmtyVIpOvNSKdhkqRSpFBptX5tUb3189jlSsARi h8+qSlfXBNYjhjU+ao/0aV0j28VvXOavgIJqAgSl3JyRGZoSItKnvMbqW3lLD0UYztIaZgR+ RinILEdt3oJDT7krFC+0dKgMQrdVSkSOehVQXHrbEYHVgiodKSiqrsDYPZQoUW/io/OBG5uC Usay1OKFmOkpUdJXleIB0WlyckIdzBPq/ynlDJRZtehM8I+U7EGzoAzmW/cQ6r6CpYRZAFZK leGJZI/OF/XolOqFujWWk/m4HQkWj1AIJfDmAhZvV73j7v//AJzLE+44tmn+AAAAAElFTkSu QmCC</item> <item item-id="13">iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtCAYAAAAN1C2uAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AZlJREFUaEPtmuESgyAMg7f3f+hNPZxYKGmVCUr8s5snFD+SgG7vz3S8eOgEZkA8dAIvwikT ICCgkGEATSFzyCzHWh0q1a7RDIeAkFWoIBC2ESCpqPV7/Ln2NoTFwl5vuWdpNfi9XTJcW1nL IQIK87CCyIV1zlrDWkzaDK1uQ2SQVEhst1JALzCvTYKtmkfqmgVK1rDelzaOphbLBaYWlt7z VjA/AJn9UVyzGwV5QaDVxwNK2ixumwAqXSyLooArDRLdoLbqoPMeMJZrd4CQH7UOc0GH4A0F yDITSH3/sJg2cdbzySpGBaVTnVgMSf+IWnJtUB2UNah9tXHuEjssedb9BcoZS2bJ+toWwHO+ FhyTxWoWu2NfcBW7403VHDMBBZrq40xN2nftqxT4zR41eobZxbNYr4ASNfU60Bbjym4lWgyk 95q0GJghAiIgn4ldL8x8XT//au6DkN2er4Ht5+Z4Gbe+iXi8guQzlhfS4wEt73TEPztW11hU REDMICqoqAGUQchmQ1jszEpNQMygM/pp+PeXc8O+rvUXLd7hprZmVL0AAAAASUVORK5C YII=</item> <item item-id="14">iVBORw0KGgoAAAANSUhEUgAAADgAAAAlCAYAAAAXzipbAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AT5JREFUaEPtmOEOgzAIhLtX34tvalZzYimHc4vX6B+tVukHBzQ+XtNRRj5mwJGPMjLcos4b UNwDdwTFA6idg1N3W/w/n/EagyIr0QoVQcoC1sjVaFXQ3X3lHHOhPtKV74NDA0Y5uOYmathW IxyfKWX0vPddC3DU/qbIWMPMQrKGWaedZfvvgLbKtRzEOoFx7mHAbzzMvsvO64Hu+qC3I2By hc2bzMIzc5tqsDezgIxMPBvMuy1AdGTvutkHs4Cega5soBFHkKdHkCkC0aKi5+yi2XmpHGQB jxr38hSV87MqGnle8flaRctzW1CjsQrsQjXDIBAzlgOsoJmzAmTJRs5K9+qQG0CMXl24Qi72 dlDSvyxsS2vueq4usWh9Uc+Uj+CaSvDrEJ0iD4iyHFqi3tZRPoJhjkYT1J/fEVSP4BuSGdUK hsIiPgAAAABJRU5ErkJggg==</item> <item item-id="15">iVBORw0KGgoAAAANSUhEUgAAAEwAAAASCAYAAADxEzisAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA APNJREFUWEftlusOwjAIhef7P7RuizW0cjuAuhia7McSoIevp2y3+762Xn4CB7BefgKbP7Qj z9vYGDACDQzjhTlsn4xg+e+EH7rG8/yITe9RFVy/EIGrAhuQKJgKrQ0MsNqAtUKbHEaDuATp 1DKnSa9Ttk6lw0xg0hzg7C4d1No8nStcDmv5xJyUmgSMdYa+uYpoYh22JmVOXhOLArN0eIF5 6kjOLwX2Dw6zxk4psIj1Nfuj9ZDxoY0VbXy8gFkzTLOxZXFL3Lp3BNTIiWrR/t+mm5MRl83N NJfdO5oP/bhGN7EcVl33k/UaGEj3p8BArZcIfwDc/tsjBfNwQQAAAABJRU5ErkJggg==</item> <item item-id="16">iVBORw0KGgoAAAANSUhEUgAAAGgAAAAlCAYAAACu2qwTAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AftJREFUaEPtWNG2wjAI8/7/R1+dx+10CARYV3nInnSlhSak0P39v54Hn74IbATx6YvAo29o jOx9uhGG3giQoB/w8yp4h9fxtxYKCbqBoA10C/hxzLPbwyJBkwlCivh0zVTQZNxDy0XIsQiy 1EQFhaDHRjs5oWNL1CCvJpEgjH3IQhLjqcklZCCPbXYI+piRJIQExXBbZkWClkFdcxQlaD8K 5TE3pUmIdiloi1qQco5lk32PYkHj2p5XxpBqEmYQFMk0yyb7HoGPxrWsjsSP1s2MLydIUwoK 2LuVy3uF9h+t740jQmYkret/HNR6ea8lvLLxfS7aICJHrhPtnqr+EWEzMDlxMgY6Slr+Rlk2 nstWwcuqxyycxrcuRLZUV9b+brVqGJ+OOEstkY1UMie6bkRFo/pRLNEEihAS3cO+lpbI1rv3 HO2Iy2aa58ACK7sxNbs+t+7ssfMrglDitFFQtq4hBWUIuuo746tCyFcZiNagTMZn65VW6EdV ajFqQCFlaHtATYW2phbbDDKggu5ywnXrCKTuQXU3nFlFgARVkbswL1MuSNAFoCtTUZ00m4SK M86pIUAF1XBbNmu8VFtt+2GzLCo6OhCQx5wk7HRXI27rEfAuu19j68OjRxLUPAdIUGOC5Gci 9KGZ96DGZG6hkSAS1ByB5uFRQc0JegJhUWGlWE8IZwAAAABJRU5ErkJggg==</item> <item item-id="17">iVBORw0KGgoAAAANSUhEUgAAAB0AAAARCAYAAAAhUad0AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AHZJREFUSEtj/A8EDPQGIEvpDRjobSE4ZIe1pcBkA0o7EEwvn4IsgwG6WArzIYalyBLoiuCK kVxLTgjBfIviU7ggAcNR4gcprohxCNY4RQ57YgwhVQ3ZllLVp8QGL6m+Q08T8DilVUJCDhW6 Zhn0kKFLPh25lgIAvZi/lsV1PpsAAAAASUVORK5CYII=</item> <item item-id="18">iVBORw0KGgoAAAANSUhEUgAAACAAAAARCAYAAAC8XK78AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AKdJREFUSEvNVAEKgCAQs/8/ulTymGtXCsolBCLXbW47jzOvFLkKgciVIsGr+r8jkONQMlG/ Ox/GsZ2tJN0pwABIZCUo9jIC3u123HqKQFcMtrBNrJZSD221fQNQN8U8KABFzssNZ6vVTVmg MsKA1pjC7Na9KfA1BWpSFJCHUVXxZFQ/eTbxhGCdt39YgICuX/A+jCjH4DK4u+Z7tO//nuJR 5qvqwhW4APqVEFQb1zbjAAAAAElFTkSuQmCC</item> <item item-id="19">iVBORw0KGgoAAAANSUhEUgAAAFsAAAAZCAYAAABeplL+AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ASpJREFUaEPtmOsOwyAIhbv3f+itNTOhROU2sEuOyf4BhQ886l7vcx1YNQQu2Fg1BI6az+Ar TUGAoY6ACPsUs0vT2w8rRmBJkAMG8ETYPDRgA3aMQKG3SYgx2bHOqGEDdAy0+uoH0HHQKtgU NKDHoItXP3rPzoCdEdODJOM9wWOqNdtTgOTzlMdSxntiFHMb7J5M9mR74nt8pDdJG6xuRKes YuK8sK15WsGt7LmkrjiJk+0FIMnFqutWGO1U//5Po/HV2IwGblTTT2HTQiwAo0nMClvl4JlA LXitnSW/m4zQ7lpAW20tjbEUM7P1gJv5WHJ/jIxQMBEYGl+NjXS4cZnh0DU7c+sB6d1JGQek Bp51B3dJvuXrCQIfH4Ft92xfuv/tBdiF/QPsQtgfSIVVbom1VpsAAAAASUVORK5CYII=</item> <item item-id="20">iVBORw0KGgoAAAANSUhEUgAAAEwAAAARCAYAAAB3h0oCAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ASFJREFUWEftVgEOwyAI7P7/6K1daoN4wKGZWRtNliUtHtwJZ1/vfW1r8Qocgq3FK7DxoSvy O41LhpwCfyvYbiqHtza/Qu9417PQPgsLxvYknbVHF9wrkhQ5i9nEzyLfk2dUIJRTYrL41R4N WsZAPi8b0Lvzs+QanajIjHAeOfQuqq/UKv81T4vr9RwJo4F1IRaR6MQiX7IOD+XXxJiaJE7k W9boNs4pSVkGq4uLhMp0VXSAqCbPl0zizqUBzf6MrwRjTolpZ0ugng7zsNgDteyFwdY5hgXr MVG247zOzfqb9tqfjKTuCGTwaISRF7Ei6U8AazxQt1oXEPIuD1fiuB2WJXX3eNZ7q26+O+nR +iPRmktjNOET9kd+Vl0aTyA8k8MHQk/v8k/NIrQAAAAASUVORK5CYII=</item> <item item-id="21">iVBORw0KGgoAAAANSUhEUgAAADEAAAARCAYAAAB0MEQqAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AKpJREFUSEvtlVsOgCAMBPH+h1YgYrDymIIaMDbxi9p2dqkuqw0zeziI2cPMDuBv0g/xkAJ2 Rd2e+ocEyyKVbsqRgxOQ4SCkFk0QKRtDoZzFWvupaQTgstjxS3Lw1Nn+jzlmyjWNISlw106k mpTuqaYZdSDOI26cdoKoLdUPDUswLU7EdWvw3RA58Fpjeq52IqhcUk5eN3rHNUNraw7/iSXw PwRR6Y2cTzixAXYKVUfKEGCBAAAAAElFTkSuQmCC</item> <item item-id="22">iVBORw0KGgoAAAANSUhEUgAAADoAAAAUCAYAAADcHS5uAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AORJREFUWEftluEOgzAIhN37P/Smxia0HnCFuZlgk/3pauG7O9le73UtFdYGWmEtFSD31D6g P1RgHRGnatte+2Rb2e/JXpJ9HsGM4EiI2bp/Bz2mftf35aCtQCQyGdU9MPm91yOKfBdd7cB0 RMS7xb5nLKjXoyVIF12rYMRlViQW1Iu51uNpGGkFkVIahFT92456oK2nERiCavbLS1in2HOz jmrCW/swukgRDzQ6jDTXrf3mLBIIujn+jmrNzkSXdTF6LiyoLFgC1BscV05d1l2vR+ueW/wz YkEz5x7QjHp3fLaMox9JUbAk7IIkWAAAAABJRU5ErkJggg==</item> <item item-id="23">iVBORw0KGgoAAAANSUhEUgAAADIAAAA9CAYAAADvaTpkAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AblJREFUaEPtmFlyxDAIRDP3P3TiqcQejRZ4YGLJCf5zCSG6G7Tw+Ny+j7/wPYHM/DYO8fKS LfeCl+OGFhC719GcaUA8ICQwU4CIKbKl2nNcA1qPLwWkCU6on+lANKbLCtNsy/HLFdGC04p6 BHRJIKRGfs6/A9elQKgaHlUSCD/2Xpb/RpG9NmiNlHWydGoR1XeVEwhhq7ax1ghZIxUhLI1s UhGBvUytTK3vJscZHrpzM7XOUJqK5K51Jn8Ae9Q9ScXli51e5ZcGcgQHtusGSPmo2R8sI1aI 5L30ofNOAam7Er3/3rW8ftVJwRIgb70qjyJNewU4oUVLuyI1UAvw5s5glTVSkZ4vDcyw2C1A ohXxvCjFXYuycHsgVgC0Rjx+1dTyONXmaEpr86Ut/Sj2+hzxONXmXAJECyJiPIGAi2j82xMs GqFuvYEkEA+rWSMgXTO1MrU8DGRqBbNmvTRarku3aD4QPpcFQttAtznZ6eE5RRHSmSnTSQPz 1nEheRhpowWXQCLZpr6IKppN0wOji0fajYL0nB/H7hUZoMWXxrjkqzf30ttvHZwHzFBNC4u/ YWsBI9lOVSSSmC+U418NF7Lz3AAAAABJRU5ErkJggg==</item> <item item-id="24">iVBORw0KGgoAAAANSUhEUgAAACsAAAAYCAYAAABjswTDAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AMVJREFUWEftld0OgCAIhev9H7q0qUNHcQDdYpPLxs93TqjnleKIEhk2ShxRQJ8N2LAGB9Kx EavkDLGFPyGDhoCtkCrYqo4Wo4o93ppgy33bzUXU0gIqeBTPCaL9kVndzmqLZzhae6hhqbtI 8QircVaT2wRxA7mVePvmcXdcIakXe3Vxrq4+bMifhGFXOyu5yj63XwoR9chQa05zVnvVWAd6 6lTP7W+cRRRvWMSlkgOvAbLTirmmVBjW1H1y0YadbGhrF8rZG6errRm8S/+JAAAAAElFTkSu QmCC</item> <item item-id="25">iVBORw0KGgoAAAANSUhEUgAAADIAAAA9CAYAAADvaTpkAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AZFJREFUaEPtWNESwiAM0///aN085zEEmlQYxcte6QppmkJ7f2zf7R++HcjMb4shvH3LFvcC b4cbMiAOr7V/pgHxgGiBmQLEAmGtv3V9oj4ckB3EEkA8h6ypLvV1OSMCUqBlGiMIGyUhtwr6 4fPS1BKQBiVhGTnKL8teyNTCHzmvB+/LXECYqFkPPo+v3KcY8UQRFTDjWxphopXbipEVL0SW cWmEjVhqL41II7/kDxA9q1FCBxBpExbqiZJrCNHUElXLBSRvaFoNDrJBKX3Y/xD7IiMstSlY pLNDDsY++aup9VkgpuRobQgLZBQjHsDFqsU4QtlA51Xp3sg5mlXLcmCtI7PaWjHIWbYCtUT5 tUCEvRCRg9eatVA3u4BoQOfJgT1qAy9ZacRDihgBmjWlllLLEwGlVueoMV0f0mmWhn6hxM62 2su8fpF7J3w/goCYxgjT7iJATm3xIF1X3SIHRIuDgPRiD2WlZfdV4XodjvFTOyAzQQkBBBU9 M1a69EKsTUC6sMk4GWGL6sVicSojPQPzBC+KfO9ji3MMAAAAAElFTkSuQmCC</item> <item item-id="26">iVBORw0KGgoAAAANSUhEUgAAACsAAAAZCAYAAACo79dmAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ALlJREFUWEftloEOhSAIRe3/P/qlLjbiSVdyNViwtWwmHi83c/vVKFGiwUaJEgW0OyBhVxQo VT/l8qdsA6UQbX+wvCoc1r1nE3blg7oam8p+Xlm+1x5i+N66RMXiwtaTYjsu9vVQWz5Le1K/ xbZabpTrT1k+QGuPwDiAhNHeP/+scJGnYNGKLaqOAGfzq7DSEgjIqixZjd/hHKhEs6tGE2n9 lvxDo1i9ehfUomp/d2Wit8cm7FOKh1J2B13yXHhZScF1AAAAAElFTkSuQmCC</item> <item item-id="27">iVBORw0KGgoAAAANSUhEUgAAACoAAAAYCAYAAACMcW/9AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AL5JREFUWEftlgEOgCAIRe3+hy5t4QhBlHTTBVtbrcTXB4TjjBZ2sAS6g4UdIO+oO6hBgVgq qV74y+Bv2pIECUbvlw09Bl06Rx10dOK6or9UFJ+l+bgarcQsf69zFP7kmahyh8DPFIQmfQso VYxTsNintrHUKTgYqfVJ4EVVo67E+m8BtajWqixES/u+aKEARdNAdSQME7V1eC/VvxRGbjjQ nFnet0aLHUp6c9MC2CvEstOTWvVf1Jm51hUdre4FXuhChMUZEhIAAAAASUVORK5CYII=</item> <item item-id="28">iVBORw0KGgoAAAANSUhEUgAAAFMAAAARCAYAAAChIpBnAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA ARBJREFUWEftVlsOgzAMY/c/9AZolbrQxM5jlI9WQmJr6sZOHPF672tbq0aBQ8y1ahTYamAW yunwJUOdAtPE3IfUhcXoP4bqca49THwfo52N5DJFTE3IEAFRFA+GjEW/UaFuF9Mi6xFCI5bB yLrlR8zeKrL9ozaSlrKqmxGi4UYxKop86cweVHvXZg6aXYgo2oc2G8xhdOb7nX3O3Gy3U2Jm SbJdk73nX+dZXFVMaXmran1HWuMAJYX2Z40INq9hbzP2ZuwjY1BSaJ+xYRSj2X10B4v5GDHR vPU4gy20dBRTLNMd7MVVcWyVq+7L4njyvf0707JTlnj1eY+QJ6/qBFg8b6IsblVcJL9pYlaR fhLOB8Nef433wb/dAAAAAElFTkSuQmCC</item> <item item-id="29">iVBORw0KGgoAAAANSUhEUgAAAFsAAABACAYAAAB1JwvBAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA Ag5JREFUeF7tmoGuwiAMRfX/P9qnM/MRpLQwdobbXWJitsEtp3eFiffH87jpYAi8YOtgCNwY GaksFUQYOAKCzbGWs0HWPuznNP1arSwfHdsIVAmugAV7G+S1dQj2GCn1YsJW2RhvDsEez9Ts UbAFGyQASsnZgg0SAKXk7DPC1lKy8qvfSDh6A30/PlgZGZk88MkfKiXYQ3HWOxNswQYJgFJy tmCDBEApxNna7YGXfj0GOmK52Kpp3V86jzi7F3TrwHt00jatL19efPn1aWEvb1wDN5mjfe15 33SwU3eVNpxb3Zc713N/Ctva8C6WiOQfCHmMn34s8WiGveBbrucDtQbe0md+rzeuHHBLDGnb UpJ3dXa6Csm/l4B91bisjNRA1bRK1yIms+Lx4rASvCvsVgdugR3Vijq7NGdYzvVK1c+UkXyA Hqwa9EjbUtlY+/RgH1pGom6zll8jJ8goaKvkeGUwnRR/YoLsSc5RbSLJy0vRVDX7KHC9uh7w rzkoMiv3BnOFdhbw0nk5G3SEYJ8Jdm2WBsc5hRTibG99OgUJIAgEdultDBjbdBKCDaYEga0y 8s7o7rA1Qf4/OrvDBp/S6aUEG0yRYAs2SACUkrMFGyQASsnZgg0SAKWKzvZ2IMD4TiUl2GA6 P7DT3WNQ/1JSgg2m21yNgDFcRkqwwVQLtmCDBEApOVuwQQKglJwNwv4DT5zD9oUZ2K8AAAAA SUVORK5CYII=</item> <item item-id="30">iVBORw0KGgoAAAANSUhEUgAAAGYAAAA2CAYAAAA1UeyVAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AfpJREFUeF7tnItuwyAMRbv//+itjZSIUQx2HuY2PZEqVSvBN/dgA0m0n9/n8eDQc+AFhkPP gYeeJBQtVQwbNB0AjCYXMkaUyxjMc7nyWrUtH448B7purzAAkwdkjeQCky+LiCYYStfcwQGY uf6b0QEDGFEHRGWRMYARdUBUFhkDmLgD37xkl82Yb7/bIAtmeSbxxffnABOvsClnACbF5niQ VDDlIwTP4wRKWQOogikKGuJj/ZwzUjMmKhkwghnjKXWl7BkQozGt9q2/S2eMN8Nm7HmiMUcQ 699vAebsPc/IxO3xr3Oftae/y8CUIyo6uvZkSuv9hCNxPWaWbaz3I5plqni5pda49WOZ4BE2 MrAUO2ob/b02xTIp2m9k3qphRDRY3qSAObvE9Ex7q9GdMlMuLDzfPYPXit8b4FY2Lb55gl45 6vb2PQJxZsZ7PPooMNmlrI53BI7n3FbpqhcGVj/TSlk5qR2ZhHsZVcdYS6c1EXuz0wulVQZL DdZ197xJm2O8Zty5nQd0PR9fPsfc2fDItY3gvM1RnoktIoC2tgOj+ebfqhMwmkOJUqbJ5fp9 jOh1y8siY0QRAQYwog6IyiJjPgnMaDMkei23ktXMGMDMZ7yBKW/IzZeFAsCIjgH+qwJgRB0Q lUXGAEbUAVFZZAxgRB0QlUXGiIL5Axbbzab7NHLWAAAAAElFTkSuQmCC</item> </binaryContent> </worksheet>