What is the speed of an Airplane flying at elevation of 6 mi on flight path over radar when distance is decreasing at 400 MPH when plane is 8 miles away from radar?

Question

Calculus - Related Rates

An Airplane is flying at an elevation of 6 miles on a flight path that will take it directly over a radar tracking station. If the distance between the plane and the station is decreasing at a rate of 400 mph when the plane is 8 miles away from being directly over the station, what is the speed of the plane?

Answer 1

This forms a right triangle where s is the hypotenuse which is the distance between the plane and the radar station. Let y be the altitude of the plane and x = the flight path of the plane.

s² = y² + x²

Given: y = 6 which is a constant

ds/dt = -400, negative since decreasing

Need to find dx/dt

Taking the implicit derivative of both sides of the above equation

2sds/dt = 0 +2xdx/dt

dx/dt = (s/x) ds/dt

When x = 8

s = √(6²+8²) = √100 = 10

dx/dt = (10/8)(-400) = -500 mph or 500 mph moving toward 6 miles above the radar station