Difference between revisions of "009A Sample Final 3, Problem 8"

Latest revision as of 15:56, 2 December 2017

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} denotes the weight in pounds of an individual, and $t$ denotes the time in months, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dW}{dt}} is the rate of weight gain or loss in lbs/mo. The current speed record for weight loss is a drop in weight from 487 pounds to 130 pounds over an eight month period. Show that the rate of weight loss exceeded 44 lbs/mo at some time during the eight month period.

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-<span class="exam">Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; and volume &nbsp;<math style="vertical-align: 0px">V</math>&nbsp; satisfy the equation &nbsp;<math style="vertical-align: 0px">PV=C</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is a constant. Suppose that at a certain instant, the volume is &nbsp;<math style="vertical-align: -4px">600 \text{ cm}^3,</math>&nbsp; the pressure is &nbsp;<math style="vertical-align: -4px">150 \text{ kPa},</math>&nbsp; and the pressure is increasing at a rate of &nbsp;<math style="vertical-align: -4px">20 \text{ kPa/min}.</math>&nbsp; At what rate is the volume decreasing at this instant?
+<span class="exam">If &nbsp;<math style="vertical-align: 0px">W</math>&nbsp; denotes the weight in pounds of an individual, and &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; denotes the time in months, then &nbsp;<math style="vertical-align: -13px">\frac{dW}{dt}</math>&nbsp; is the rate of weight gain or loss in lbs/mo. The current speed record for weight loss is a drop in weight from 487 pounds to 130 pounds over an eight month period. Show that the rate of weight loss exceeded 44 lbs/mo at some time during the eight month period.
+<hr>
+[[009A Sample Final 3, Problem 8 Solution|'''<u>Solution</u>''']]
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Foundations: &nbsp;
-|-
-|'''Product Rule'''
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)</math>
-|}
+[[009A Sample Final 3, Problem 8 Detailed Solution|'''<u>Detailed Solution</u>''']]
-'''Solution:'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we take the derivative of the equation &nbsp;<math style="vertical-align: 0px">PV=C.</math>
-|-
-|Using the product rule, we get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>P'V+PV'=C'.</math>
-|-
-|Since &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is a constant, &nbsp;<math style="vertical-align: -1px">C'=0.</math>&nbsp;
-|-
-|Therefore, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>P'V+PV'=0.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Solving for &nbsp;<math style="vertical-align: -4px">V',</math>&nbsp; we get
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>V'=\frac{-P'V}{P}.</math>
-|-
-|Using the information provided in the problem, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: 0px">V=600 \text{ cm}^3,~P=150 \text{ kPa},~P'=20 \text{ kPa/min}.</math>&nbsp;
-|-
-|Hence, we get
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{V'} & = & \displaystyle{\frac{-(20)(600)}{150} \text{ cm}^3\text{/min}}\\
-&&\\
-& = & \displaystyle{-80 \text{ cm}^3\text{/min}.}
-\end{array}</math>
-|-
-|Therefore, the volume is decreasing at a rate of &nbsp;<math style="vertical-align: -5px">80 \text{ cm}^3\text{/min}</math>&nbsp; at this instant.
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;The volume is decreasing at a rate of &nbsp;<math style="vertical-align: -5px">80 \text{ cm}^3\text{/min}</math>&nbsp; at this instant.
-|}
 [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Final 3, Problem 8"

Latest revision as of 15:56, 2 December 2017

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