Car A is traveling west at 40 mi/h and car B is traveling north at 60 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?â
EXAMPLE 4 Car A is traveling west at 40 mi/h and car B is traveling north at 60 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 to the left, where C is the intersection of the roads. 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 dydt =-40 mi/h and dy/dt =-60 mi/h. (The derivatives are negative because x, y, and z are decreasing.) We are asked to find dIdt. The equation that relates x, y, and z is given by the Pythagorean Theorem: Video Example) Tutorial Online Textbook (Enter differentials such as dy/dx as (dy)/(dx).) Differentiating each side with respect to t, we have dt dz 1 When x-0.4 mi and y = 0.3 mi, the Pythagorean Theorem give z = 0.5 mi, so dz 0.4 +0.3 dt mi/h The cars are approaching each other at a rate of mi/h