Question

A gas station stands at the intersection of a north-south road and an east-west road. A police car is traveling towards the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph when it is 3 miles from the gas station. At the same time the truck is 4 miles from the gas station going 80 mph. At this moment:
a. Is the distance between the car and the truck increasing or decreasing? How fast?
b. Repeat part a) if the truck is going 70 mph instead of 80 mph

Answer 1

a = distance apart = (x^2 + y^2)^.5
at present a = 5 miles (3,4,5 triangle)

da/dt = .5 (x^2+y^2)^-.5 (2x dx/dt + 2 y dy/dt)
if dx/dt = -100
dy/dt = +80
then
da/dt = (.5/5)(2*3*-100 + 2*4*80)
= (.1)(-600 + 640)
= +4
the truck is making a getaway at the moment.
You can do it for the truck doing 70 mph. I suspect the truck will be losing then.

Answer 2

SO B is -4. which is decreasng