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

Answer:

The minimum cost is $17.30c

Step-by-step 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

$z = \frac{288}{xy}$

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 $z = \frac{288}{xy}$

$C = 0.06 * 2(y*\frac{288}{xy} + x*\frac{288}{xy}) + 0.16 * xy$

$C = 0.06 * 2(\frac{288}{x} + \frac{288}{y}) + 0.16 * xy$

$C = 0.12(\frac{288}{x} + \frac{288}{y}) + 0.16 * xy$

$C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 * xy$

$C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 xy$

Differentiate w.r.t x and y

$C_x = -\frac{34.56}{x^2} + 0.16y$

$C_y = -\frac{34.56}{y^2} + 0.16x$

By comparison: $x = y$

Set them equal to 0

$C_y = -\frac{34.56}{y^2} + 0.16x=0$

$-\frac{34.56}{y^2} + 0.16x=0$

Substitute x for y

$-\frac{34.56}{x^2} + 0.16x=0$

$0.16x=\frac{34.56}{x^2}$

Cross multiply

$0.16x^3 = 34.56$

$x^3 = \frac{34.56}{0.16}$

$x^3 = 216$

Take the cube root of both sides

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

$x = 6$

$x=y= 6$

Substitute 6 for x and for y in $C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 xy$

$C = (\frac{34.56}{6} + \frac{34.56}{6}) + 0.16 * 6* 6$

$C = (\frac{2*34.56}{6}) + 5.76$

$C = 11.52 + 5.76$

$C = 17.28$

$C = 17.3$ --- approximated