A home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 218.75 cubic feet. The cost of wall paint is $0.04 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. sides ft height ft What is the minimum cost for the paint? (Round your answer to two decimal places.)

The dimensions of the room will be: Length: 5 ft, Width: 5 ft, Heigth: 8.75 ft.

The cost of the paint is $10.5.

Step-by-step explanation:

We have a room, with a volume of 218.75 cubic feet.

For a optimized room, the sides of the wall will be equal, as the cost of painting a wall are equal. This means we will have a square ceiling.

Then we have to write the cost function in function of the unit cost of the paint and the surface of walls and ceiling:

- We have four walls of surface

- We have one ceiling with surface

Then, the cost function is:

As the volume is a constraint, we can write z in function of x as:

Replacing in the cost function, we have:

To optimize the cost function, we derive and equal to zero

The height of the ceiling will be:

The dimensions of the room will be

Length: 5 ft, Width: 5 ft, Heigth: 8.75 ft.

The cost of the painting will be

the answer is ! ?