Difference between revisions of "007A Sample Midterm 2, Problem 5"

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(Created page with "<span class="exam"> A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the s...")
 
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<span class="exam"> A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?
 
<span class="exam"> A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?
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!Solution: &nbsp;  
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|'''The Pythagorean Theorem'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; For a right triangle with side lengths &nbsp;<math style="vertical-align: -4px">a,b,c</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; is the length of the
 
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&nbsp; &nbsp; &nbsp; &nbsp; hypotenuse, we have &nbsp;<math style="vertical-align: -2px">a^2+b^2=c^2.</math>
 
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!Exam Solution: &nbsp;  
 
 
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|[[File:9AF_5_GP.png|center|550px]]
 
|[[File:9AF_5_GP.png|center|550px]]
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'''Detailed Solution:'''
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'''Detailed Solution'''
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!Foundations: &nbsp;
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|'''The Pythagorean Theorem'''
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|&nbsp; &nbsp; &nbsp; &nbsp; For a right triangle with side lengths &nbsp;<math style="vertical-align: -4px">a,b,c</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; is the length of the
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&nbsp; &nbsp; &nbsp; &nbsp; hypotenuse, we have &nbsp;<math style="vertical-align: -2px">a^2+b^2=c^2.</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 16:35, 2 November 2017

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?

Solution:  
550px
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 30^2+h^2=s^2}
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=50}

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{h} & = & \displaystyle{\sqrt{s^2-30^2}}\\ &&\\ & = & \displaystyle{\sqrt{50^2-30^2}}\\ &&\\ & = & \displaystyle{40} \end{array}}

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 2hh'=2ss'}
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 2(40)6=2(50)s'}
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{s'} & = & \displaystyle{\frac{2(40)(6)}{2(50)}}\\ &&\\ & = & \displaystyle{\frac{24}{5} \text{ m/s}} \end{array}}


Detailed Solution

Foundations:  
The Pythagorean Theorem
        For a right triangle with side lengths    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 the length of the

        hypotenuse, 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 a^2+b^2=c^2.}


Step 1:  
550px
From the diagram, 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 30^2+h^2=s^2}   by the Pythagorean Theorem.
Taking derivatives, 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 2hh'=2ss'.}

Step 2:  
If   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=50,}   then
       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 h=\sqrt{50^2-30^2}=40.}
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 2(40)6=2(50)s'.}
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 s',}   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 s'=\frac{24}{5} \text{ m/s.}}  


Final Answer:  
       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'=\frac{24}{5} \text{ m/s}}  

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