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

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|To graph <math>\rho(x),</math> we need to find out when <math>-x^2+6x+16</math> is negative.
 +
|-
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|To do this, we set
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>-x^2+6x+16=0.</math>
 +
|-
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|So, we have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 +
\displaystyle{0} & = & \displaystyle{-x^2+6x+16}\\
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&&\\
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& = & \displaystyle{-(x^2-6x-16)}\\
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&&\\
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& = & \displaystyle{-(x+2)(x-8).}
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\end{array}</math>
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|-
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|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>
 +
|-
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|Using test points, we can see that <math>-x^2+6x+16</math> is positive in the interval <math>[0,8]</math>
 +
|-
 +
|and negative in the interval <math>[8,12].</math>
 
|-
 
|-
−
|
+
|Hence, we have
 
|-
 
|-
−
|
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|<math>\rho(x) = \left\{
 +
    \begin{array}{lr}
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      -x^2+6x+16 &  \text{if }0\le x \le 8\\
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      x^2-6x-16 & \text{if }8<x\le 12
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    \end{array}
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  \right.
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</math>
 
|-
 
|-
−
|
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|The graph of <math>\rho(x)</math> is displayed below.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
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|We need to find the absolute maximum and minimum of <math>\rho(x).</math>
 +
|-
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|We begin by finding the critical points of <math>-x^2+6x+16.</math>
 +
|-
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|Taking the derivative, we have <math>-2x+6.</math>
 +
|-
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|Solving <math>-2x+6=0,</math> we get a critical point at <math>x=3</math>.
 +
|-
 +
|Now, we calculate <math>\rho(0),\rho(3),\rho(12).</math>
 
|-
 
|-
−
|  
+
|We have
 
|-
 
|-
−
|
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|<math>\rho(0)=16,\rho(3)=25,\rho(12)=56.</math>
 
|-
 
|-
−
|
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|Therefore, the minimum of <math>\rho(x)</math> is <math>16</math> and the maximum of <math>\rho(x)</math> is <math>56.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)'''  
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|&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)
 
|-
 
|-
 
|'''(b)'''  
 
|'''(b)'''  

Revision as of 12:13, 3 March 2017

The population density of trout in a stream is

where    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:  
Step 2:  


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 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.} (See Step 1 for graph)
(b)

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