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

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!Step 1:    
 
!Step 1:    
 
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|-
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|First, we take the derivative of the equation &nbsp;<math style="vertical-align: 0px">PV=C.</math>
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|-
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|Using the product rule, we get
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>P'V+PV'=C'.</math>
 
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|Since &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is a constant, &nbsp;<math style="vertical-align: -1px">C'=0.</math>&nbsp;
 
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|-
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|Therefore, we have
 
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>P'V+PV'=0.</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
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|-
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|Solving for &nbsp;<math style="vertical-align: -4px">V',</math>&nbsp; we get
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>V'=\frac{-P'V}{P}.</math>
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|-
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|Using the information provided in the problem, we have
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|-
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|&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;
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|-
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|Hence, we get
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{V'} & = & \displaystyle{\frac{-(20)(600)}{150} \text{ cm}^3\text{/min}}\\
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&&\\
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& = & \displaystyle{-80 \text{ cm}^3\text{/min}.}
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\end{array}</math>
 
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|-
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|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.
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&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.
 
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|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 10:57, 7 March 2017

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure    and volume  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 V}   satisfy the equation  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 PV=C}   where  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 C}   is a constant. Suppose that at a certain instant, the volume is  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 600 \text{ cm}^3,}   the pressure is  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 150 \text{ kPa},}   and the pressure is increasing at a rate of  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 20 \text{ kPa/min}.}   At what rate is the volume decreasing at this instant?

Foundations:  
Product Rule
        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{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)}


Solution:

Step 1:  
First, we take the derivative of the equation  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 PV=C.}
Using the product rule, we get
       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 P'V+PV'=C'.}
Since  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 C}   is a constant,  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 C'=0.}  
Therefore, we have
       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 P'V+PV'=0.}
Step 2:  
Solving for  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 V',}   we get
       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 V'=\frac{-P'V}{P}.}
Using the information provided in the problem, we have
       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 V=600 \text{ cm}^3,~P=150 \text{ kPa},~P'=20 \text{ kPa/min}.}  
Hence, we get

        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 \begin{array}{rcl} \displaystyle{V'} & = & \displaystyle{\frac{-(20)(600)}{150} \text{ cm}^3\text{/min}}\\ &&\\ & = & \displaystyle{-80 \text{ cm}^3\text{/min}.} \end{array}}

Therefore, the volume is decreasing at a rate of  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 80 \text{ cm}^3\text{/min}}   at this instant.


Final Answer:  
       The volume is decreasing at a rate of  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 80 \text{ cm}^3\text{/min}}   at this instant.

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