Difference between revisions of "009B Sample Final 3, Problem 3"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|To graph <math>\rho(x),</math> we need to find out when <math>-x^2+6x+16</math> is negative.
+
|To graph &nbsp;<math>\rho(x),</math>&nbsp; we need to find out when &nbsp;<math>-x^2+6x+16</math>&nbsp; is negative.
 
|-
 
|-
 
|To do this, we set  
 
|To do this, we set  
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
|Hence, we get <math>x=-2</math> and <math>x=8.</math> But, <math>x=-2</math> is outside of the domain of <math>\rho(x).</math>
+
|Hence, we get &nbsp;<math>x=-2</math>&nbsp; and &nbsp;<math>x=8.</math>&nbsp;
 
|-
 
|-
−
|Using test points, we can see that <math>-x^2+6x+16</math> is positive in the interval <math>[0,8]</math>  
+
|But, &nbsp;<math>x=-2</math>&nbsp; is outside of the domain of &nbsp;<math>\rho(x).</math>
 
|-
 
|-
−
|and negative in the interval <math>[8,12].</math>
+
|Using test points, we can see that &nbsp;<math>-x^2+6x+16</math>&nbsp; is positive in the interval &nbsp;<math>[0,8]</math>
 +
|-
 +
|and negative in the interval &nbsp;<math>[8,12].</math>
 
|-
 
|-
 
|Hence, we have
 
|Hence, we have
 
|-
 
|-
−
|<math>\rho(x) = \left\{
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\rho(x) = \left\{
 
     \begin{array}{lr}
 
     \begin{array}{lr}
 
       -x^2+6x+16 &  \text{if }0\le x \le 8\\
 
       -x^2+6x+16 &  \text{if }0\le x \le 8\\
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</math>
 
</math>
 
|-
 
|-
−
|The graph of <math>\rho(x)</math> is displayed below.
+
|The graph of &nbsp;<math>\rho(x)</math>&nbsp; is displayed below.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|We need to find the absolute maximum and minimum of <math>\rho(x).</math>
+
|We need to find the absolute maximum and minimum of &nbsp;<math>\rho(x).</math>
 
|-
 
|-
−
|We begin by finding the critical points of <math>-x^2+6x+16.</math>
+
|We begin by finding the critical points of &nbsp;<math>-x^2+6x+16.</math>
 
|-
 
|-
−
|Taking the derivative, we have <math>-2x+6.</math>
+
|Taking the derivative, we have &nbsp;<math>-2x+6.</math>
 
|-
 
|-
−
|Solving <math>-2x+6=0,</math> we get a critical point at <math>x=3</math>.
+
|Solving &nbsp;<math>-2x+6=0,</math>&nbsp; we get a critical point at &nbsp;<math>x=3</math>.
 
|-
 
|-
−
|Now, we calculate <math>\rho(0),\rho(3),\rho(12).</math>
+
|Now, we calculate &nbsp;<math>\rho(0),\rho(3),\rho(12).</math>
 
|-
 
|-
 
|We have
 
|We have
 
|-
 
|-
−
|<math>\rho(0)=16,\rho(3)=25,\rho(12)=56.</math>
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\rho(0)=16,\rho(3)=25,\rho(12)=56.</math>
 
|-
 
|-
−
|Therefore, the minimum of <math>\rho(x)</math> is <math>16</math> and the maximum of <math>\rho(x)</math> is <math>56.</math>
+
|Therefore, the minimum of &nbsp;<math>\rho(x)</math>&nbsp; is &nbsp;<math>16</math>&nbsp; and the maximum of &nbsp;<math>\rho(x)</math>&nbsp; is &nbsp;<math>56.</math>
 
|}
 
|}
  
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|To calculate the total number of trout, we need to find
 
|To calculate the total number of trout, we need to find
 
|-
 
|-
−
|<math> \int_0^{12} \rho(x)~dx.</math>
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math> \int_0^{12} \rho(x)~dx.</math>
 
|-
 
|-
 
|Using the information from Step 1 of (a), we have
 
|Using the information from Step 1 of (a), we have
 
|-
 
|-
−
|<math> \int_0^{12} \rho(x)~dx.=\int_0^8 (-x^2+6x+16)~dx+\int_8^{12} (x^2-6x-16)~dx.</math>
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math> \int_0^{12} \rho(x)~dx.=\int_0^8 (-x^2+6x+16)~dx+\int_8^{12} (x^2-6x-16)~dx.</math>
 
|}
 
|}
  
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\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
|So there are approximately <math>251</math> trout.
+
|Thus, there are approximately &nbsp;<math>251</math>&nbsp; trout.
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; The minimum of <math>\rho(x)</math> is <math>16</math> and the maximum of <math>\rho(x)</math> is <math>56.</math> (See Step 1 for graph)
+
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; The minimum of &nbsp;<math>\rho(x)</math>&nbsp; is &nbsp;<math>16</math>&nbsp; and the maximum of &nbsp;<math>\rho(x)</math>&nbsp; is &nbsp;<math>56.</math>&nbsp; (See Step 1 for graph)
 
|-
 
|-
−
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; There are approximately <math>251</math> trout.
+
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; There are approximately &nbsp;<math>251</math>&nbsp; trout.
 
|-
 
|-
 
|  
 
|  
 
|}
 
|}
 
[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 12:28, 3 March 2017

The population density of trout in a stream is

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 \rho}   is measured in trout per mile and  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 x}   is measured in miles.  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 x}   runs from 0 to 12.

(a) Graph  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 \rho(x)}   and find the minimum and maximum.

(b) Find the total number of trout in the stream.

Foundations:  
What is the relationship between population density  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 \rho(x)}   and the total populations?
       The total population is equal 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 \int_a^b \rho(x)~dx}
       for appropriate choices 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 a,b.}


Solution:

(a)

Step 1:  
To graph  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 \rho(x),}   we need to find 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 -x^2+6x+16}   is negative.
To do this, 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 -x^2+6x+16=0.}
So, 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 \begin{array}{rcl} \displaystyle{0} & = & \displaystyle{-x^2+6x+16}\\ &&\\ & = & \displaystyle{-(x^2-6x-16)}\\ &&\\ & = & \displaystyle{-(x+2)(x-8).} \end{array}}
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 x=-2}   and  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 x=8.}  
But,  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 x=-2}   is outside of the domain 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 \rho(x).}
Using test points, we can 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 -x^2+6x+16}   is positive 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 [0,8]}
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 [8,12].}
Hence, 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 \rho(x) = \left\{ \begin{array}{lr} -x^2+6x+16 & \text{if }0\le x \le 8\\ x^2-6x-16 & \text{if }8<x\le 12 \end{array} \right. }
The graph 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 \rho(x)}   is displayed below.
Step 2:  
We need to find the absolute maximum and minimum 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 \rho(x).}
We begin by finding the critical points 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 -x^2+6x+16.}
Taking the derivative, 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 -2x+6.}
Solving  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 -2x+6=0,}   we get a critical point at  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 x=3} .
Now, 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 \rho(0),\rho(3),\rho(12).}
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 \rho(0)=16,\rho(3)=25,\rho(12)=56.}
Therefore, the minimum 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 \rho(x)}   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 16}   and the maximum 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 \rho(x)}   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 56.}

(b)

Step 1:  
To calculate the total number of trout, we need to find
       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^{12} \rho(x)~dx.}
Using the information from Step 1 of (a), 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 \int_0^{12} \rho(x)~dx.=\int_0^8 (-x^2+6x+16)~dx+\int_8^{12} (x^2-6x-16)~dx.}
Step 2:  
We integrate to 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^{12} \rho(x)~dx} & = & \displaystyle{\bigg(\frac{-x^3}{3}+3x^2+16x\bigg)\bigg|_0^8+\bigg(\frac{x^3}{3}-3x^2-16x\bigg)\bigg|_8^{12}}\\ &&\\ & = & \displaystyle{\bigg(\frac{-8^3}{3}+3(8)^2+16(8)\bigg)-0+\bigg(\frac{(12)^3}{3}-3(12)^2-16(12)\bigg)-\bigg(\frac{8^3}{3}-3(8)^2-16(8)\bigg)}\\ &&\\ & = & \displaystyle{8\bigg(\frac{56}{3}\bigg)+12\bigg(\frac{12}{3}\bigg)+8\bigg(\frac{56}{3}\bigg)}\\ &&\\ & = & \displaystyle{\frac{752}{3}.} \end{array}}
Thus, there are approximately  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 251}   trout.


Final Answer:  
    (a)     The minimum 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 \rho(x)}   is    and the maximum 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 \rho(x)}   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 56.}   (See Step 1 for graph)
    (b)     There are approximately  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 251}   trout.

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