Calculus word plane problem

Question

Two planes approach an air traffic control tower. One is flying due east at 650 mi/hr and the other is flying due south at 580 mi/hr. Assuming that they fly at the same constant elevation, how fast is the distance between planes changing when east bound plane is at 400 miles from the tower and the south bound is 300 miles from the tower?

Answer 1

Let x = distance of the eastbound plane from the tower at time t

y = distance of the southbound plane from the tower at time t

z = distance between the planes at time t

x, y, and z form a right triangle with legs x and y.

Given: dx/dt = -650 and dy/dt = -580

Find: dz/dt when x = 400 and y = 300

By the Pythagorean Theorem, z² = x² + y²

When x = 400 and y = 300, z² = 400² + 300². So, z = 500.

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

dz/dt = [x(dx/dt) + y(dy/dt)] / z = [400(-650) + 300(-580)] / 500 = -868

The planes are approaching each other at the rate of 868 miles per hour.