Question

You are flying a kite at a constant height of 150 ft. The wind blows the kite horizontally to the left at a rate of 25 ft/min. At what rate is the length of the string changing when it is stretched out to 200 ft

Answer 1

Answer:

The length of the string changing when it is stretched out to 200 ft at rate 16.535 ft/min

Step-by-step explanation:

Refer the attached figure

Height of kite = y = 150 feet

We are given that  The wind blows the kite horizontally to the left at a rate of 25 ft/min. i.e. $\frac{dx}{dt}=25$

We are supposed to find At what rate is the length of the string changing when it is stretched out to 200 ft

z= 200

Using Pythagoras theorem :

$x^2+y^2=z^2 ---1\\x^2+(150)^2=(200)^2\\x=\sqrt{(200)^2-(150)^2}\\x=132.28$

Now to find the rate at which the length of the string changing when it is stretched out to 200 ft

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt}\\2(132.28)(25)+2(150)(0)=2(200)\frac{dz}{dt}\\6614=2(200)\frac{dz}{dt}\\\frac{6614}{2(200)}=\frac{dz}{dt}\\16.535=\frac{dz}{dt}$

Hence The length of the string changing when it is stretched out to 200 ft at rate 16.535 ft/min