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

to plot points   for x just start at the origin (middle) and go to the left however many times it say(if the number is a positive) if the number is a negative then go right however many times it says if it is 0 go nowhere

now keep you pencil there and go up however many times it says on y(if it is positive) if it is negative then go down however many times it says. if it says 0 go nowhere.

you have just plotted a point! now when you finish plotting all of them connect them with a ruler

plot each point as (-3, 7), (0,4), (3,1), (8,-4)   in the parenthesis on the left of the comma is x and the right is y.