Question

Two cars are moving away from an intersection along roads that make a 90 degree angle. The first car is 100 feet from the intersection and is travelling at a rate of 30 ft/s. The second care is 150 feet away from the intersection and is travelling at a rate of 40ft/s. Find the rate of change of the distance between the cars after one second.

Answer 1

d = sqrt(x^2 + y^2); where x = 100 ft, y = 150 ft, dx/dt = 30ft/s, dy/dt = 40ft/s
After one second, x = 130 ft, y = 190 ft

ds/dt = (2x dx/dt + 2y dy/dt)/(2sqrt(x^2 + y^2) = (2(130)(30) + 2(190)(40)) / 2sqrt(130^2 + 190^2) = (7,800 + 15,200) / 2sqrt(16,900 + 36,100) = 23,000 / 2sqrt(53,000) = 49.95 ft/s

Therefore, the rate of change of the distance between the cars after one second is approximately 50 ft/s.