Question

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?

Answer 1

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