Please solve this question

Question

A young child is flying a kite which is at height of 50 m. The wind is carrying the kite horizontally away from the child at the speed of 6.5 m/s . How fast must the kite string be let out when the string is 130 m.

Answer 1

This is a related rate. You have the height and the horizontal speed given. The distance of the kite from the child is going to be equal to the hypotenuse of a right triangle with one leg of the right triangle equal to 50m^2 and the other leg growing at a rate of 6m/s.

How fast then is the hypotenuse growing? First, find the length of the hypotenuse given the two sides of the triangle. (Note: hypot = z, horizontal = x, height = y)

z^2=x^2+y^2

Take the derivative with respect to time and get:

2z(dz/dt)=2x(dx/dt)+2y(dy/dt)

Plug in all the known variables.

We know that dy/dt = 0 because the height is constant. dx/dt is given as 6.5 and we have to find dz/dt which is our answer since recall dz/dt is the rate at which the string is changing with respect to time.

z is given as 135. x is $\sqrt{135^2-50^2}=120$ and y is 50.

Thus, the answer is $\frac{dz}{dt}=\frac{2\cdot 120\cdot 6.5}{135\cdot 2}$ Simplify and get $5.\bar{7}$meters/second. That is the rate at which the length of string is increasing at the moment the string is 135 meters long.