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:

Question

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