# 007A Sample Midterm 2, Problem 5 Detailed Solution

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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?

Background Information:
The Pythagorean Theorem
For a right triangle with side lengths  ${\displaystyle a,b,c}$  where  ${\displaystyle c}$  is the length of the

hypotenuse, we have  ${\displaystyle a^{2}+b^{2}=c^{2}.}$

Step 1:
From the diagram, we have  ${\displaystyle 30^{2}+h^{2}=s^{2}}$  by the Pythagorean Theorem.
Taking derivatives, we get

${\displaystyle 2hh'=2ss'.}$

Step 2:
If   ${\displaystyle s=50,}$  then
${\displaystyle h={\sqrt {50^{2}-30^{2}}}=40.}$
So, we have
${\displaystyle 2(40)6=2(50)s'.}$
Solving for   ${\displaystyle s',}$  we get   ${\displaystyle s'={\frac {24}{5}}{\text{ m/s.}}}$

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