Question

Max-Min problem please help
A box with a square bottom and no top is to be made to contain 100 cubic inches. Bottom material costs five cents per square inch and side material costs two cents per square inch. Find the cost of the least expensive box that can be made.

Answer 1

Will become fan of anyone who helps me out :(

Answer 2

I need to be able to provide a full proof of my answer, and I havent any idea as to where to start

Answer 3

Does no one know how to help me....

Answer 4

h = height, w = width
Since this is a rectangular prism:
\[V = Volume = hw^2 = 100\]Solve for h:\[h=\frac{100}{w^2}\]C = cost = 5(Area of bottom) * 2(Area of other 4 sides)
\[C=5(w^2)+2(4wh)=5w^2+8wh\]Replace h with the expression we found above.\[C=5w^2+8w \frac{100}{w^2}=5w^2+\frac{800}{w}\]To find a max/mins of C, take the derivative with respect to w and set it equal to 0(the slope of a horizontal tangent).\[\frac{dC}{dw}=10w-\frac{800}{w^2}=0\]\[10w=\frac{800}{w^2}\]\[w^3=80\]\[w = \sqrt[3]{80}\]So the least expensive box has a width of about 4.31 inches (cube root of 80)and costs about $278.50 (substitute w into equation for cost).