An open-topped glass aquarium with a square base is designed to hold 13.513.513, point, 5 cubic feet of water. what is the minimum exterior surface area of the aquarium?
Accepted Solution
A:
Suppose the aquarium's shape is a square prism, with each side of the square having length x, and the aquarium's height is y, so we have: x^2 * y = 13.5 y = 13.5/(x^2) The area of each side is xy, and there are four sides, so that totals 4xy The area of the bottom is x^2 Total surface area: A = x^2 + 4xy A = x^2 + 4x(13.5/(x^2)) A = x^2 + 54x^(-1) dA/dx = 2x - 54x^(-2) Set the derivative to zero: 2x - 54x^(-2) = 0 2x = 54/(x^2) 2x^3 = 54 x^3 = 54/2 x^3 = 27 x = cbrt(27) = 3, assuming x is a real number A = 3^2 + 54*3^(-1) A = 9 + 18 A = 27 square feet