Question

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.

Answer 1

The minimum cost is $17.30c

Explanation:

Given

`V = 288` --- Volume

`C_1 = 0.06` --- cost of wall paint

`C_2 = 0.16` ---cost of ceiling paint

Required

Minimum cost of paint

The volume is calculated as:

`V =xyz`

Substitute 288 for V

`288 =xyz`

Make z the subject

`z = (288)/(xy)`

The surface area is calculated as:

`Area = 2(yz + xz) + xy`

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:

`C = 0.06 * 2(yz + xz) + 0.16 * xy`

Substitute
`z = (288)/(xy)`

`C = 0.06 * 2(y*(288)/(xy) + x*(288)/(xy)) + 0.16 * xy`

`C = 0.06 * 2((288)/(x) + (288)/(y)) + 0.16 * xy`

`C = 0.12((288)/(x) + (288)/(y)) + 0.16 * xy`

`C = ((34.56)/(x) + (34.56)/(y)) + 0.16 * xy`

`C = ((34.56)/(x) + (34.56)/(y)) + 0.16 xy`

Differentiate w.r.t x and y

`C_x = -(34.56)/(x^2) + 0.16y`

`C_y = -(34.56)/(y^2) + 0.16x`

By comparison:
`x = y`

Set them equal to 0

`C_y = -(34.56)/(y^2) + 0.16x=0`

`-(34.56)/(y^2) + 0.16x=0\\`

Substitute x for y

`-(34.56)/(x^2) + 0.16x=0`

`0.16x=(34.56)/(x^2)`

Cross multiply

`0.16x^3 = 34.56`

`x^3 = (34.56)/(0.16)`

`x^3 = 216`

Take the cube root of both sides

`x = \sqrt[3]{216}`

`x = 6`

`x=y= 6`

Substitute 6 for x and for y in
`C = ((34.56)/(x) + (34.56)/(y)) + 0.16 xy`

`C = ((34.56)/(6) + (34.56)/(6)) + 0.16 * 6* 6`

`C = ((2*34.56)/(6)) + 5.76`

`C = 11.52 + 5.76`

`C = 17.28`

`C = 17.3` --- approximated