Posted by **hanan** on Sunday, November 6, 2011 at 8:49pm.

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**, Sunday, November 6, 2011 at 9:22pm
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(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.

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