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 $0.11 per square foot. Find the room dimensions that result in a minimum cost for the paint.
I will begin to start you off. Step one, lets draw a diagram. We can think of this as a cube with an open top (top meaning in this case the floor thats not being painted). Step 2 would to begin relating some equations and defining variables. Let x equal widths and y be heights or vica versa (its your preference when defining these) Thirdly, We know volume=668.25 cm^3 which was given and the equation for the volume of a cube is going to be V=4xy+x^2 and then we have an equation to solve in terms of x and y.
668.25=4xy+x^2 and we also know for max area covered the equation will be x^2*y.
^^ with the equation above you can solve for the dimensions in cm^3 of x (width) and y (height) of the rectangular room. Now we can relate these dimensions to a surface area equation for a rectangular room (which is a cube with an open top) meaning the floor isn’t painted. Lastly, dealing with the constraints of cost. Quite simply once you solve for the minimum dimensions in sq ft multiply the the cost for your x dimensions by 0.06$ 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!