A 10-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at 3 meters per minute. At a certain instant, the bottom of the ladder is 6 meters from the wall. What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)? Choose 1 answer: (A) -(7)/(2) (B) 7 (C) 12 (D) -6

Question

A 10-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at 3 meters per minute. At a certain instant, the bottom of the ladder is 6 meters from the wall. What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)? Choose 1 answer: (A)  -(7)/(2) (B) 7 (C) 12 (D) -6

Bytelearn · Accepted Answer

Denote Variables:  Let's denote the distance from the top of the ladder to the ground as y (in meters). The ladder's length is 10 meters. We're given that y is decreasing at 3 meters per minute, so dy/dt = -3 m/min. Calculate Triangle Area:  The area of the right-angled triangle formed by the wall, ground, and ladder can be expressed as A = (1/2) * base * height. Here, the base is the distance from the wall, which is 6 meters, and the height is y. So, A = (1/2) * 6 * y. Find Rate of Change:  To find the rate of change of the area, we need to differentiate A with respect to time (t). So, dA/dt = (1/2) * 6 * dy/dt. We substitute dy/dt with -3 to get dA/dt = (1/2) * 6 * (-3). Check Right-Angled Triangle:  Calculating dA/dt gives us dA/dt = (1/2) * 6 * (-3) = -9 square meters per minute. But wait, we didn't use the ladder's length to ensure the triangle is right-angled. We need to check if the ladder's length and the distances given form a right-angled triangle.

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