Question

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.)

Answer 1

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.

Explanation:

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

`V=x\cdot y \cdot z = 218.75`

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.

`x=y`

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
`S_w=xz`

- We have one ceiling with surface
`S_c=x^2`

Then, the cost function is:

`C=0.04*4*xz+0.14x^2=0.16xz+0.14x^2`

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

`V=x^2z=218.75\\\\z=(218.75)/(x^2)`

Replacing in the cost function, we have:

`C=0.16xz+0.14x^2\\\\C=0.16x((218.75)/(x^2))+0.14x^2 =(35)/(x)+0.14x^2`

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

`C =(35)/(x)+0.14x^2\\\\(dC)/(dx)=(35*(-1))/(x^2) +0.14*2x\\\\(dC)/(dx)=-(35)/(x^2)+0.28x=0\\\\0.28x=35x^(-2)\\\\x^3=35/0.28= 125\\\\x=\sqrt[3]{125} =5`

The height of the ceiling will be:

`z=(218.75)/(x^2) =(218.75)/(5^2) =(218.75)/(25) =8.75`

The dimensions of the room will be

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

The cost of the painting will be

`C=0.16xz+0.14x^2\\\\C=0.16*(5*8.75)+0.14*(5^2)\\\\C=7+3.5=10.5`