Question

Two cars, car X and car Y , start moving from the same point P on a cross intersection. Car X is travelling east and car Y is travelling north. Some time later car X is 60 km east of point P and travelling in an easterly direction at 80 km/h and car Y is 80 km north of point P and travelling in a northerly direction at 100 km/h. How fast is the distance between car X and car Y changing?

Answer 1

let the distance traveled by car X be x km
let the distance traveled by car Y by y km
their paths form a right-angled triangle.
Let the distance between them be D km
D^2 = x^2 + y^2
2D dD/dt = 2x dx/dt + 2y dy/dt
dD/dt = (x dx/dt + y dy/dt)/D

at the given case:
x = 60, y = 80 , dx/dt = 80, dy/dt = 100
D^2 = 60^2 + 80^2 = 10000
D = 100

dD/dt = (60(80) + 80(100))/100
= 128