Question

A car is traveling north towards an intersection at 60mph at the same time a truck is headed east toward the same intersection at 45mph. Find the rate of change of the distance between the car and truck when the car is 3 miles south of the intersection and the truck is 4 miles west of the intersection. Is this distance increasing or decreasing at that time?

Answer 1

Place the car on the negative y axis and label the distance to the origin y
Place the truck on the negative x axis and label the distance to the origin x
label the distance between them d
then
d^2 = x^2 + y^2
2d dd/dt = 2x dx/dt + 2y dy/dt
dd/dt =(x dx/dt + y dy/dt)/d

Given: when x = 4 , dx/dt = -45
when y = 3 , dy/dt = -60
d = 5 , (did you notice the 3-4-5 right-angled triangle ?)

dd/dt = (4(-45) + 3(-60)/5 = -72

At that moment the distance between them is <b>decreasing</b> at 72 mph

The distance between them is decreasing, since dd/dt is negative.

Answer 2

12