The base of an decorative rectangular feature with a volume of V = 225 ft^3 is to be constructed in a restaurant. The bottom is made of marble, the sides are made of glass, and the top is open. If marble costs five times as much (per unit area) as glass, find the dimensions of the feature that minimize the cost of the materials. (Assume that the length is greater than or equal to the width. Give your answers correct to at least three decimal places.)
The dimensions are x,y,z so we have z = 225/(xy)
Further , let's say we have y=x since that gives minimum perimeter to the base, and hence minimum cost for the glass sides.
If glass costs 1, then marble costs 5, and the cost function
c(x) = 5x^2 + 4xz = 5x^2+4*225/x^2
= 5(x^2+180/x^2)
dc/dx = 10(x-180/x^3) = 10/x^3 (x^4-180)
dc/dx=0 when x^4 = 180