The perimeter of a square is increasing at a rate of 5 meters per hour. At a certain instant, the perimeter is 30 meters. What is the rate of change of the area of the square at that instant (in square meters per hour)? Choose 1 answer: (A) (5)/(4) (B) 25 (C) (75)/(4) (D) (25)/(16)

Question

The perimeter of a square is increasing at a rate of 5 meters per hour. At a certain instant, the perimeter is 30 meters. What is the rate of change of the area of the square at that instant (in square meters per hour)? Choose 1 answer: (A)  (5)/(4) (B) 25 (C)  (75)/(4) (D)  (25)/(16)

Bytelearn · Accepted Answer

Calculate Perimeter Length: First, let's find the length of one side of the square using the perimeter. The perimeter (P) of a square is 4 times the length of one side (s), so P = 4s.Solve for Side Length: Given P = 30 meters, we can solve for s: 30 = 4s, so s = 30 / 4 = 7.5 meters.Find Area: Now, we know the area (A) of a square is s^2. So, A = (7.5)^2 = 56.25 square meters.Perimeter Rate of Change: The rate of change of the perimeter is 5 meters per hour. Since the perimeter is 4 times the length of one side, the rate of change of the length of one side is 5 / 4 meters per hour.Area Rate of Change: To find the rate of change of the area, we use the derivative of A with respect to s, which is dA/ds = 2s. Then we multiply this by the rate of change of s, which is ds/dt.Final Rate of Change: So, the rate of change of the area is dA/dt = dA/ds * ds/dt = 2s * (5/4).Plugging in the value of s, we get dA/dt = 2 * 7.5 * (5/4) = 15 * (5/4) = 18.75 square meters per hour.

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