Difference between revisions of "009A Sample Final A, Problem 9"

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<span style="font-size:135%"> <font face=Times Roman>  9. A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of &nbsp; <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>.
 
<span style="font-size:135%"> <font face=Times Roman>  9. A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of &nbsp; <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>.
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! Foundations: &nbsp;  
 
! Foundations: &nbsp;  
 
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|Like most geometric word problems, you should start with a picture.  This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate. In particular, we need to choose variables to describe the distance between the bug and the point (3,4), which we can call ''z''.  By the given information, we can consider the position on the ''x''-axis simply as ''x''.
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|Like most geometric word problems, you should start with a picture.  This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate. In particular, we need to choose variables to describe the distance between the bug and the point <math style="vertical-align: -21%;">(3,4)</math>, which we can call <math style="vertical-align: 0%;">z</math>.  By the given information, we can consider the position on the <math style="vertical-align: 0%;">x</math>-axis simply as <math style="vertical-align: 0%;">x</math>.
 
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'''Solution:'''
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&nbsp;'''Solution:'''
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|'''Write the Basic Equation:''' From the picture, we can see there is a triangle involving both the bug and the point (3,4).  From this, we can see that &zwnj; <math style="vertical-align: -7%;">z^2 = x^2 +4^2.</math>
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|'''Write the Basic Equation:''' From the picture, we can see there is a right triangle involving both the bug and the point <math style="vertical-align: -21%;">(3,4)</math>.  From this, we can see that <math style="vertical-align: -8%;">z^2=x^2+4^2</math>.
 
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!Step 3: &nbsp;  
 
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|'''Evaluate and Solve:''' When the bug is at the origin, we have ''x'' = 3.  By the Pythagorean Theorem, ''z'' = 5.  Based on our drawing, ''x'' is actually ''decreasing'' at a rate of 30, so we should really write <math style="vertical-align: -19%;">dx/dt = -30</math>. We now simply plug in to the result of our implicit differentiation to find
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|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -22%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
 
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|&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>
 
|&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>

Latest revision as of 19:41, 27 May 2015

BugGP.png

9. A bug is crawling along the 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} -axis at a constant speed 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 \frac{dx}{dt}=30} . How fast is the distance between the bug and the point 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 (3,4)} changing when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).

Foundations:  
Like most geometric word problems, you should start with a picture. This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate. In particular, we need to choose variables to describe the distance between the bug and the point 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 (3,4)} , which we can call 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 z} . By the given information, we can consider the position on the 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} -axis simply as 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} .

 Solution:

Step 1:  
Write the Basic Equation: From the picture, we can see there is a right triangle involving both the bug and the point 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 (3,4)} . From this, 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 z^2=x^2+4^2} .
Step 2:  
Use Implicit Differentiation: We take the derivative of the equation from Step 1 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 2z\frac{dz}{dt} = 2x\frac{dx}{dt},}
or
     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{dz}{dt} = \frac{x}{z}\cdot\frac{dx}{dt},}
Step 3:  
Evaluate and Solve: When the bug is at the origin, 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 x=3} . By the Pythagorean Theorem, 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 z=5} . Based on our drawing, our variable 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 actually 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 30} , so we should really write 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 dx/dt=-30} . We now simply plug in to the result of our implicit differentiation 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 \frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.}

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