related rates problem?

Question

A police helicopter is flying north at 100 mph, and at constant altitude .5 miles. Below, a car is traveling west on a hgihway at 75 mph. At the moment the helicopter crosses over the highway, the care is 2 miles east of the helicopter.
a) How fast is the distance between the car and the helicopter changing at the moment the helicopter crosses the highway?
b) Is the distance between the car and helicopter increasing or decreasing at that moment?

Answer 1

As the helicopter approaches the highway, its speed contributes to a decreasing distance between car and helicopter. After the helicopter crosses the highway, its speed contributes to an increasing distance between car and helicopter. At the instant it is over the highway, its speed does not contribute to an increase or decrease in the distance between car and helicopter. The change in distance is only affected at that instant by the speed of the car.

You have a triangle with height 1/2 mile and side 2 miles.

Let
x = distance between car and point on ground below helicopter
z = distance between car and helicopter

Given dx/dt = -75 miles/hr

Find dz/dt when x = 2 miles.
_____________

z² = (1/2)² + x² = 1/4 + x²
z² = 1/4 + 2² = 1/4 + 4 = 17/4
z = √(17/4) = √17 / 2
when x = 2

2z(dz/dx) = 2x
z(dz/dx) = x

dz/dx = x/z = 2/(√17/2) = 4/√17
when x = 2

dz/dt = (dz/dx)(dx/dt) = (4/√17)(-75) = -300/√17
dz/dt ≈ -72.76 miles/hr