posted by hanan .
A cardboard box without a top is to have volume 500000 cubic cm. Find the dimensions which minimize the amount of material used. List them in ascending order.
You can assume symmetry between length and width, which reduces to the width(=length) and the height.
Furthermore, one of the two can be eliminated from the volume relation:
So the minimization problem is reduced to one single dimension as in elementary calculus.
The same results can be obtained by calculating the area of material required:
A=2h(b+w)+bw + L(bwh-500000)
the second term introduces the Lagrange multiplier.
Take partial derivatives with respect to w,b,h and L and solve for each variable from the 4 equations.
This method should give the same results as in part A.