The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers. What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)? Choose 1 answer: (A) -204 (B) -148 (C) 148 (D) 204 The surface area of a square prism with base side s and height h is 2s^(2)+4sh.

Question

The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers. What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)? Choose 1 answer: (A) -204 (B) -148 (C) 148 (D) 204 The surface area of a square prism with base side  s and height  h is  2s^(2)+4sh.

Bytelearn · Accepted Answer

Surface Area Formula: The surface area (SA) of a square prism is given by the formula SA = 2s^2 + 4sh, where s is the side of the base and h is the height.Differentiation with Respect to Time: To find the rate of change of the surface area, we need to differentiate the surface area formula with respect to time (t). So we get d(SA)/dt = d(2s^2)/dt + d(4sh)/dt.Product Rule Application: Using the product rule for differentiation, the rate of change of the first term 2s^2 with respect to time is d(2s^2)/dt = 4s * ds/dt, since the derivative of s^2 with respect to s is 2s and then we multiply by ds/dt (the rate of change of s).Rate of Change Calculation: Similarly, the rate of change of the second term 4sh with respect to time is d(4sh)/dt = 4s * dh/dt + 4h * ds/dt, using the product rule (derivative of the first times the second plus the first times the derivative of the second).Given Rates and Values: Now we plug in the rates given in the problem: ds/dt = -7 km/min (since the side is decreasing) and dh/dt = 10 km/min (since the height is increasing).Calculation of First Term Rate: We also plug in the values for s and h at the instant given: s = 4 km and h = 9 km.Calculation of Second Term Rate: So the rate of change of the first term 2s^2 is 4s * ds/dt = 4 * 4 km * (-7 km/min) = -112 km^2/min.Total Rate of Change Calculation: And the rate of change of the second term 4sh is 4s * dh/dt + 4h * ds/dt = 4 * 4 km * 10 km/min + 4 * 9 km * (-7 km/min) = 160 km^2/min - 252 km^2/min.Adding these two rates of change together gives us the total rate of change of the surface area: d(SA)/dt = -112 km^2/min + (160 km^2/min - 252 km^2/min).Calculating the sum gives us d(SA)/dt = -112 km^2/min - 92 km^2/min = -204 km^2/min.

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