GAIA Car A is traveling west at 40 mi/h and car B is traveling north at 70 mi/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.4 mi and car B is 0.3 mi from the intersection? Solution We draw the figure below, where C is the intersection of the roads. U At a given time t, let x be the distance from car A to C, let y be the distance from car B to C, and let z be the distance between the cars, where x, y, and z are measured in miles. We are given that X = -46 mi/h and y = -70 mi/h. (The derivatives are negative because x and y are decreasing.) We are asked to find . The equation that relates x, y, and z is given by the Pythagorean Theorem. 2 2 = x2 + 12 Differentiating each side with respect to t, we have dz 2z_ = 2x x + 2y- dt dt dz (x_dx dt Z dt dt When x = 0.4 mi and y = 0.3 mi, the Pythagorean Theorem gives z = 0.5 mi, so dz 1 ( )(-40) + ()(-70) 11 mi/h . The cars are approaching each other at a rate of mi/h