Calculus (Related Rates)

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.



An illustration shows a police car and a truck, both are moving towards Sioux Falls. The interconnecting path shows a right angled triangle. The car is moving downward at 160 kilometers per hour toward Sioux Falls which is at a distance y. A truck is moving at 140 kilometers per hour at a distance of x, away from Sioux Falls.

At time 𝑡=0, the police car is 50km north and the truck is 40km east of Sioux Falls.

Calculate the rate at which the distance between the vehicles is changing at 𝑡=12 minutes.

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

Answer 1

dy/dt = -160 km/h

dx/dt = 140 km/h

distance between the vehicles, d: d² = x²+y²

Take the derivative 2d dd/dt = 2x dx/dt + 2y dy/dt

dd/dt = (x dx/dt + y dy/dt)/d

After 12 minutes, the police car has traveled (12/60)160 = 32 km, so it is y= 50-32 = 22 km from Sioux Falls.

After 12 minutes the truck has traveled (12/60)140 = 28 km, so it is x= 40+28 = 68 km from Sioux Falls

d = √(22²+68²) = √(484+4624) = 71.47 km apart

dd/dt = (68(140) +22(-160))/71.47 = 6000/71.47 = 83.951 km/h

Answer 2