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)?

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Find Surface Area: First, let's find the expression for the surface area of a square prism. A square prism has 2 square bases and 4 rectangular sides. The surface area (SA) is given by: SA = 2 * (side length)^2 + 4 * (side length) * heightDifferentiate Surface Area: Now, let's differentiate the surface area with respect to time to find the rate of change of the surface area. We'll use the product rule for differentiation where necessary. d(SA)/dt = 2 * 2 * (side length) * d(side length)/dt + 4 * [d(side length)/dt * height + (side length) * d(height)/dt]Given Rates: We are given the rates of change of the side length and the height: d(side length)/dt = -7 km/min (since the side is decreasing) d(height)/dt = 10 km/min (since the height is increasing)Substitute Values: Now we substitute the given values and the rates into the differentiated surface area formula: d(SA)/dt = 2 * 2 * 4 km * (-7 km/min) + 4 * [-7 km/min * 9 km + 4 km * 10 km/min]Perform Calculations: Let's perform the calculations: d(SA)/dt = 2 * 2 * 4 * (-7) + 4 * [-7 * 9 + 4 * 10] d(SA)/dt = 16 * (-7) + 4 * [-63 + 40] d(SA)/dt = -112 + 4 * (-23) d(SA)/dt = -112 - 92 d(SA)/dt = -204 square kilometers per minute

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