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

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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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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Solution: &nbsp;
 
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|-
 
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|[[File:9AF_5_GP.png|center|550px]]
 
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|-
 
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|<math style="vertical-align: -3px">30^2+h^2=s^2</math>
 
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|<math style="vertical-align: -4px">s=50</math>
 
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|
 
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<math>\begin{array}{rcl}
 
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\displaystyle{h} & = & \displaystyle{\sqrt{s^2-30^2}}\\
 
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&&\\
 
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& = & \displaystyle{\sqrt{50^2-30^2}}\\
 
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&&\\
 
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& = & \displaystyle{40}
 
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\end{array}</math>
 
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|<math>2hh'=2ss'</math>
 
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|<math>2(40)6=2(50)s'</math>
 
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|<math>\begin{array}{rcl}
 
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\displaystyle{s'} & = & \displaystyle{\frac{2(40)(6)}{2(50)}}\\
 
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&&\\
 
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& = & \displaystyle{\frac{24}{5} \text{ m/s}}
 
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\end{array}</math>
 
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|}
 
  
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(insert picture of handwritten solution)
  
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'''Detailed Solution'''
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[[007A Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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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;"
 
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!Step 1: &nbsp;
 
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|[[File:9AF_5_GP.png|center|550px]]
 
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|From the diagram, we have &nbsp;<math style="vertical-align: -3px">30^2+h^2=s^2</math>&nbsp; by the Pythagorean Theorem.
 
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|Taking derivatives, we get
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>2hh'=2ss'.</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|If &nbsp; <math style="vertical-align: -4px">s=50,</math>&nbsp; then
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -2px">h=\sqrt{50^2-30^2}=40.</math>
 
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|So, we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -5px">2(40)6=2(50)s'.</math>
 
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|Solving for &nbsp; <math style="vertical-align: -5px">s',</math>&nbsp;  we get &nbsp; <math style="vertical-align: -14px">s'=\frac{24}{5} \text{ m/s.}</math> &nbsp;
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -14px">s'=\frac{24}{5} \text{ m/s}</math>&nbsp;
 
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|}
 
 
[[007A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[007A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 06:25, 3 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?

(insert picture of handwritten solution)

Detailed Solution for this Problem

Return to Sample Exam

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