How fast must she let out the string when the kite is 500 ft away from her

Question

A girl flies a kite at a height of 300 feet, the wind carrying the kite horizontally away from her at a rate of 25 feet per second.

Answer 1

Let x = distance of girl from the point on the ground directly below the              kite at time t

 

     y = length of string at time t

 

At time t, we have a right triangle with horizontal leg of length x, vertical leg of length 300, and hypotenuse of length y.

 

Given:  dx/dt = 25

 

Find:  dy/dt when y = 500

 

Solution:  By the Pythagorean Theorem, x² + 300² = y²

 

                                                       2x(dx/dt) = 2y(dy/dt)

 

              Since x² + 300² = y² and y = 500, x = 400

 

              So, 2(400)(25) = 2(500)(dy/dt)

 

              dy/dt = 20 ft/sec