Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 640 cubic feet. The cost of wall paint is $0.076 per Square foot and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width, And height of a rectangular room respectively. Find the room dimensions that result in a minimum Cost for the paint. Round your answers to two decimal places.
A) $x = y = 8.00$ and $z = 10.00$
B) $x = y = 10.00$ and $z = 12.80$
C) $y = z = 10.00$ and $x = 12.80$
D) $x = z = 8.00$ and $y = 10.00$
E) $y = z = 8.00$ and $x = 10.00$

Question

Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 640 cubic feet. The cost of wall paint is $0.076 per Square foot and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width, And height of a rectangular room respectively. Find the room dimensions that result in a minimum Cost for the paint. Round your answers to two decimal places. 
A) $x = y = 8.00$ and $z = 10.00$
B) $x = y = 10.00$ and $z = 12.80$
C) $y = z = 10.00$ and $x = 12.80$
D) $x = z = 8.00$ and $y = 10.00$
E) $y = z = 8.00$ and $x = 10.00$

Quizplus · Accepted Answer

The cost of painting is minimized when the surface area to be painted is minimized. Given the volume of the room is fixed at 640 cubic feet, the dimensions that minimize the surface area (and thus the cost) are those of a cube or as close to a cube as possible. Among the given options, $x = y = 8.00$ and $z = 10.00$ are the dimensions that most closely approach a cube, leading to a minimized surface area for painting.

Suppose a Home Improvement Contractor Is Painting the Walls and Ceiling