If the length is x and the width is y, and the height is z, then
xyz = 50, so z = 50/(xy)
The area is
a(x,y) = 2(xz+yz+xy)
and so the cost is
c(x,y) = 2(300xz+550yz+0.02xy)
= 2((300x+550y)(50/(xy))+0.02xy)
Now, normally, I'd set ∂c/∂x=0 and ∂c/∂y = 0 and solve for x and y, but when I do that here, I get a ridiculously small value for z, since the cost of the sides is so much greater than that of the bottom. You sure there are no typos?
the jewelry box will have rectangular sides and an open top. the longer sides will be made of gold at a cost of $300/in^2 and the shorter sides will be made from platinum at a price of $550/in^2. the bottom of the box will be made from plywood at a cost of $.02/in^2. what dimensions will provide me with the lowest cost if i would like the box to have a volume of 50in^3?
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