Related rates (Car moving towards intersection)

Question

Suppose car A is heading south at 100 km/h and car B is heading east at 90 km/h. Both are headed for the intersection at point C. At what rate are the cars

approaching each other when car A is 64 m and car B 61 m from the intersection? Enter your answer accurate to 2 decimal places.

Answer 1

Let x = distance of car A from point C at time t

y = distance of car B from point C at time t

z = distance between the cars at time t

Given: dx/dt = -100 and dy/dt = -90

Find: dz/dt when x = 64 and y = 61

x² + y² = z²

When x = 64 and y = 61, z = √[64² + 61²] = √7817

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

dz/dt = [x(dx/dt) + y(dy/dt)] / z = -11890/√7817 = -134.48

The cars are approaching each other at the rate of 134.48 mph when car A is 64 miles from the intersection and car B is 61 miles from the intersection.