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.

The minimum cost is $17.30c

Step-by-step explanation:

Given

--- Volume

--- cost of wall paint

---cost of ceiling paint

Required

Minimum cost of paint

The volume is calculated as:

Substitute 288 for V

Make z the subject

The surface area is calculated as:

Because xy represent the dimension of the ceiling and the opposite of the ceiling (the floor) will not be painted. Hence, it does not require a coefficient of 2

The cost is:

Substitute

Differentiate w.r.t x and y

By comparison:

Set them equal to 0

Substitute x for y

Cross multiply

Take the cube root of both sides

Substitute 6 for x and for y in

--- approximated

it's