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

Answer:

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.

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

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:

$V=x^2z=218.75\\\\z=\frac{218.75}{x^2}$

Replacing in the cost function, we have:

$C=0.16xz+0.14x^2\\\\C=0.16x(\frac{218.75}{x^2})+0.14x^2 =\frac{35}{x}+0.14x^2$

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

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

The height of the ceiling will be:

$z=\frac{218.75}{x^2} =\frac{218.75}{5^2} =\frac{218.75}{25} =8.75$

The dimensions of the room will be

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

The cost of the painting will be

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

Answer 2

Answer:

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: