SOLUTION: 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 and side material costs two cents per square inch. Find

Let x be the size of the squared base edge and h be the height.


Then the volume is  x^2*h = 100  cubic inches 

and the lateral surface area is 4*xh square inches.


The total cost is  C(x,y) = 5x^2 + 2*4*xh = 5x^2 + 8xh.


So we need to minimize  C(x,y) = 5x^2 + 8xh  under restriction x^2*h = 100.

We then express h =  and substitute it into the expression for C(x,y).


We then get

    C(x,y) =  +  =  + ,

and we need to minimize this function.


Find the derivative and equate it to zero

    10x -  = 0.


From this equation, find x

    10*x^3 = 800

    x^3 =  = 80.

    x =  = 4.309  inches.

Then h =  =  = 5.386 inches.


ANSWER.  The base size is 4.309 inches;  the height is 5.386 inches.