Difference between revisions of "009B Sample Midterm 2, Problem 3"

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&nbsp; &nbsp; They are related by the equation <math style="vertical-align: -5px">v(t)=s'(t).</math>
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&nbsp; &nbsp; &nbsp; &nbsp; They are related by the equation <math style="vertical-align: -5px">v(t)=s'(t).</math>
 
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|'''2.''' If we calculate <math style="vertical-align: -14px">\int_a^b v(t)~dt,</math> what are we calculating?
 
|'''2.''' If we calculate <math style="vertical-align: -14px">\int_a^b v(t)~dt,</math> what are we calculating?
 
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&nbsp; &nbsp; We are calculating <math style="vertical-align: -5px">s(b)-s(a).</math>  
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&nbsp; &nbsp; &nbsp; &nbsp; We are calculating <math style="vertical-align: -5px">s(b)-s(a).</math>  
 
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&nbsp; &nbsp; This is the displacement of the particle from <math style="vertical-align: 0px">t=a</math> to <math style="vertical-align: 0px">t=b.</math>  
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&nbsp; &nbsp; &nbsp; &nbsp; This is the displacement of the particle from <math style="vertical-align: 0px">t=a</math> to <math style="vertical-align: 0px">t=b.</math>  
 
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|'''3.''' If we calculate <math style="vertical-align: -14px">\int_a^b |v(t)|~dt,</math> what are we calculating?
 
|'''3.''' If we calculate <math style="vertical-align: -14px">\int_a^b |v(t)|~dt,</math> what are we calculating?
 
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&nbsp; &nbsp; We are calculating the total distance traveled by the particle from <math style="vertical-align: 0px">t=a</math> to <math style="vertical-align: 0px">t=b.</math>
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&nbsp; &nbsp; &nbsp; &nbsp; We are calculating the total distance traveled by the particle from <math style="vertical-align: 0px">t=a</math> to <math style="vertical-align: 0px">t=b.</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|To calculate the total distance the particle traveled from <math style="vertical-align: -1px">t=0</math> to <math style="vertical-align: -5px">t=10,</math> we need to calculate
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|To calculate the total distance the particle traveled from <math style="vertical-align: -1px">t=0</math> to <math style="vertical-align: -5px">t=10,</math>  
 
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|&nbsp; &nbsp; <math>\int_0^{10} |v(t)|~dt=\int_0^{10} |-32t+200|~dt.</math>
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|we need to calculate
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\int_0^{10} |v(t)|~dt=\int_0^{10} |-32t+200|~dt.</math>
 
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&nbsp; &nbsp; <math>\begin{array}{rcl}
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
\displaystyle{\int_0^{10} |-32t+200|~dt} & = & \displaystyle{\int_0^{6.25} -32t+200~dt+\int_{6.25}^{10}-(-32t+200)~dt}\\
 
\displaystyle{\int_0^{10} |-32t+200|~dt} & = & \displaystyle{\int_0^{6.25} -32t+200~dt+\int_{6.25}^{10}-(-32t+200)~dt}\\
 
&&\\
 
&&\\
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|The particle travels <math style="vertical-align: -1px">850</math> feet.
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|&nbsp; &nbsp; &nbsp; &nbsp; The particle travels <math style="vertical-align: -1px">850</math> feet.
 
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[[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 18:55, 7 February 2017

A particle moves along a straight line with velocity given by:

feet per second. Determine the total distance traveled by the particle

from time to time


Foundations:  
1. How are the velocity function 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(t)} and the position function 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 s(t)} related?

        They are related by 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 v(t)=s'(t).}

2. If we calculate 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 \int_a^b v(t)~dt,} what are we calculating?

        We are calculating 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 s(b)-s(a).}

        This is the displacement of the particle from 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 t=a} to 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 t=b.}

3. If we calculate 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 \int_a^b |v(t)|~dt,} what are we calculating?

        We are calculating the total distance traveled by the particle from 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 t=a} to 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 t=b.}


Solution:

Step 1:  
To calculate the total distance the particle traveled from 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 t=0} to 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 t=10,}
we need to calculate
        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 \int_0^{10} |v(t)|~dt=\int_0^{10} |-32t+200|~dt.}
Step 2:  
We need to figure out when 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 -32t+200} is positive and negative in the interval 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 [0,10].}
We set 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 -32t+200=0} and solve 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 t.}
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 t=6.25.}
Then, we use test points to see that 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 -32t+200} is positive from 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 [0,6.25]}
and negative from 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 [6.25,10].}
Step 3:  
Therefore, 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{\int_0^{10} |-32t+200|~dt} & = & \displaystyle{\int_0^{6.25} -32t+200~dt+\int_{6.25}^{10}-(-32t+200)~dt}\\ &&\\ & = & \displaystyle{\left. (-16t^2+200t)\right|_{0}^{6.25}+\left. (16t^2-200t)\right|_{6.25}^{10}}\\ &&\\ & = & \displaystyle{-16(6.25)^2+200(6.25)+(16(10)^2-200(10))-(16(6.25)^2-200(6.25))}\\ &&\\ & = & \displaystyle{850}.\\ \end{array}}


Final Answer:  
        The particle travels 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 850} feet.

Return to Sample Exam

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