Calculate the rate at which the distance between the vehicles is changing at t=5 minutes.

Question

A police car traveling south toward Sioux Falls, Iowa, at 160km/h pursues a truck traveling east away from Sioux Falls at 140km/h.

At time 𝑡=0, the police car is 40km north and the truck is 30km east of Sioux Falls.

Calculate the rate at which the distance between the vehicles is changing at t=5 minutes.

(Use decimal notation. Give your answer to three decimal places.)

Answer 1

Car equation moving along y axis: y = 40 - (160/60)t with t in min and y in km

Truck equation moving along x axis x = 30 + (140/60)t

distance between them is d = sqrt(x²+y²)

To find rates, take implicit derivatives with time

dd/dt = (1/2)d^-1/2(2x*dx/dt + 2y*dy/dt)

Evaluate at t = 5. (You know the rates of x and y, and you can calculate x, y, and d at 5 minutes)