Question

A home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 875.00 cubic feet. The cost of wall paint is $0.08 per square foot and the cost of ceiling paint is $0.14 per square foot. Find the room dimensions that result in a minimum cost for the paint.

Answer 1

Answer:

The room dimensions for a minimum cost are: sides of 10 feet and height of 8.75 feet.

Step-by-step explanation:

We have a rectangular room with sides x and height y.

The volume of the room is 875 cubic feet, and can be expressed as:

$V=x^2y=875$

With this equation we can define y in function of x as:

$x^2y=875\\\\y=\dfrac{875}{x^2}$

The cost of wall paint is $0.08 per square foot. We have 4 walls which have an area Aw:

$A_w=xy=x\cdot \dfrac{875}{x^2}=\dfrac{875}{x}$

The cost of ceiling paint is $0.14 per square foot. We have only one ceiling with an area:

$A_c=x^2$

We can express the total cost of painting as:

$C=0.08\cdot (4\cdot A_w)+0.14\cdot A_c\\\\C=0.08\cdot (4\cdot \dfrac{875}{x})+0.14\cdot x^2\\\\\\C=\dfrac{280}{x}+0.14x^2$

To calculate the minimum cost, we derive this function C and equal to zero:

$\dfrac{dC}{dx}=280(-1)\dfrac{1}{x^2}+0.14(2x)=0\\\\\\-\dfrac{280}{x^2}+0.28x=0\\\\\\0.28x=\dfrac{280}{x^2}\\\\\\x^3=\dfrac{280}{0.28}=1000\\\\\\x=\sqrt[3]{1000} =10$

The sides of the room have to be x=10 feet.

The height can be calculated as:

$y=875/x^2=875/(10^2)=875/100=8.75$

The room will have sides of 10 feet and a height of 8.75 feet.