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.
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