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

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

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.
`(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=√((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(dx)/(dt)+2y(dy)/(dt)=2z(dz)/(dt)\\2(132.28)(25)+2(150)(0)=2(200)(dz)/(dt)\\6614=2(200)(dz)/(dt)\\(6614)/(2(200))=(dz)/(dt)\\16.535=(dz)/(dt)`

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