C x A y B Video Example EXAMPLE 4 Car A is traveling west at 30...

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C x A y B Video Example EXAMPLE 4 Car A is traveling west at 30 mi/h and car B is traveling north at 50 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 0.3 mi and car B is 0.4 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 dx/dt = -30 mi/h and dy/dt = -50 mi/h. (The derivatives are negative because x and y are decreasing.) We are asked to find dz/dt. The equation that relates x, y, and z is given by the Pythagorean Theorem: z = x + y Differentiating each side with respect to t, we have 2zdz dt dz dt 2x dx dt +2ydy dt dy 1(x dx + yok). When x = 0.3 mi and y = 0.4 mi, the Pythagorean Theorem gives z = 0.5 mi, so dz dt 1 (-30) + mi/h. The cars are approaching each other at a rate of (-50)] mi/h. C x A y B Video Example EXAMPLE 4 Car A is traveling west at 30 mi/h and car B is traveling north at 50 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 0.3 mi and car B is 0.4 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 dx/dt = -30 mi/h and dy/dt = -50 mi/h. (The derivatives are negative because x and y are decreasing.) We are asked to find dz/dt. The equation that relates x, y, and z is given by the Pythagorean Theorem: z = x + y Differentiating each side with respect to t, we have 2zdz dt dz dt 2x dx dt +2ydy dt dy 1(x dx + yok). When x = 0.3 mi and y = 0.4 mi, the Pythagorean Theorem gives z = 0.5 mi, so dz dt 1 (-30) + mi/h. The cars are approaching each other at a rate of (-50)] mi/h.