Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 288 cubic feet. The cost of wall paint is $0.06 per square foot and the cost of ceiling paint is $0.16 per square foot. Let x, y, and z be the length, width, and height of a rectangular room respectively. Identify the room dimensions that result in a minimum cost for the paint and use these dimensions to find the minimum cost for the paint. Round your answer to the nearest cent.
A) $30.72
B) $7.68
C) $9.72
D) $17.28
E) $12.80

Question

Quizplus · Accepted Answer

To minimize the cost, the room should be a cube, as this minimizes the surface area for a given volume. For a cube, x = y = z, and x^3 = 288, so x = 6.63 feet. The surface area is 2(xy + xz + yz) = 2(3x^2) = 3x^2 for walls and x^2 for the ceiling. The cost is 3x^2 * $0.06 + x^2 * $0.16 = 0.34x^2. Substituting x = 6.63, the cost is approximately $17.28.

Suppose a Home Improvement Contractor Is Painting the Walls and Ceiling