Posted by hanan on .
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.

calc 3 
MathMate,
A.
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:
w^2h=500000
So the minimization problem is reduced to one single dimension as in elementary calculus.
B.
The same results can be obtained by calculating the area of material required:
A=2h(b+w)+bw + L(bwh500000)
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.