A person stands 12 meters east of an intersection and watches a car driving away from the intersection to the north at 4 meters per second. At a certain instant, the car is 9 meters from the intersection. What is the rate of change of the distance between the car and the person at that instant (in meters per second)? Choose 1 answer: (A) (20)/(3) (B) 15 (C) 4sqrt10 (D) 2.4

Question

A person stands 12 meters east of an intersection and watches a car driving away from the intersection to the north at 4 meters per second. At a certain instant, the car is 9 meters from the intersection. What is the rate of change of the distance between the car and the person at that instant (in meters per second)? Choose 1 answer: (A)  (20)/(3) (B) 15 (C)  4sqrt10 (D) 2.4

Bytelearn · Accepted Answer

Find Hypotenuse Length: First, let's find the length of the hypotenuse using the Pythagorean theorem: hypotenuse^2 = 9^2 + 12^2. That's hypotenuse^2 = 81 + 144. So, hypotenuse^2 = 225. Taking the square root, hypotenuse = 15 meters.Calculate Rate of Change: Now, let's use the formula for the rate of change of the hypotenuse (dh/dt) in a right triangle, which is given by dh/dt = (dx/dt * x + dy/dt * y) / h, where x and y are the legs of the triangle and dx/dt and dy/dt are their rates of change. Here, dx/dt = 0 (because the person is not moving east or west), dy/dt = 4 m/s (the speed of the car going north), x = 12 meters, y = 9 meters, and h = 15 meters.Apply Formula: Plugging in the values, we get dh/dt = (0 * 12 + 4 * 9) / 15. That simplifies to dh/dt = (0 + 36) / 15. So, dh/dt = 36 / 15.Simplify Result: Now, let's simplify 36 / 15. That's dh/dt = 2.4 meters per second.

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