Related Rates calculus

Question

A police car is parked 30 feet due north of a stop sign on a street road a red car is traveling towards the stop sign from 150 feet due east on the road the police radar read that the distance between the police car and the red car is decreasing at a rate of 85 ft./s how fast does a red car actually traveling along the road

Answer 1

Let

p = 30 ft be the distance of the police car from the stop sign,

c(t) be the distance of the red car from the stop sign at time t,

r(t) be the distance of the red car from the police car at time t,

t₁ be the point in time when

c(t₁) = 150 ft.

We are given

dr/dt(t₁) = 85 ft/s

and we are to calculate dc/dt(t₁).

By Pythagorean theorem at each time point t

r² = c² + p²

Differentiate the equation with respect to t:

2r dr/dt = 2c dc/dt

Thus

dc/dt = dr/dt r/c = dr/dt √(c² + p²)/c

Substitute the known values at time t₁

dc/dt(t₁) = 85×√(150² + 30²)/150 = 86.7 ft/s