009A Sample Final 1, Problem 5

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?

Foundations:
Recall:
The Pythagorean Theorem:
For a right triangle with side lengths $a,b,c$, where $c$ is the length of the
hypotenuse, we have $a^{2}+b^{2}=c^{2}.$

Solution:

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

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

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

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