**A home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 668.25 cubic feet. The cost of wall paint is $0.06 per square foot and the cost of ceiling paint is $0.11 per square foot. Find the room dimensions that result in a minimum cost for the paint.**

**Answer:**

I will begin to start you off. Step one, lets draw a diagram. We can think of this as a cube with an open top (top meaning in this case the floor thats not being painted). Step 2 would to begin relating some equations and defining variables. Let x equal widths and y be heights or vica versa (its your preference when defining these) Thirdly, We know volume=668.25 cm^3 which was given and the equation for the volume of a cube is going to be V=4xy+x^2 and then we have an equation to solve in terms of x and y.

668.25=4xy+x^2 and we also know for max area covered the equation will be x^2*y.

^^ with the equation above you can solve for the dimensions in cm^3 of x (width) and y (height) of the rectangular room. Now we can relate these dimensions to a surface area equation for a rectangular room (which is a cube with an open top) meaning the floor isnâ€™t painted. Lastly, dealing with the constraints of cost. Quite simply once you solve for the minimum dimensions in sq ft multiply the the cost for your x dimensions by 0.06$ per sq ft and then y dimensions by 0.11$ per sq ft and solve the equation once more. This shall yield the room dimensions that minimize cost for paint. Hope this helps! good luck!