# Math Help - volume optimization

1. ## volume optimization

How do I go about doing this problem:

A box with a square base and open top must have a volume of 32000 cm3. Find the dimensions of the box that minimize the amount of material used.
1 cm (length and width)
2 cm (height)

Thanks!

2. Originally Posted by skeltonjoe
How do I go about doing this problem:

A box with a square base and open top must have a volume of 32000 cm3. Find the dimensions of the box that minimize the amount of material used.
1 cm (length and width)
2 cm (height)

Thanks!
Let s = the side of the base

Let h = the height of the box

The area of the base is $s^2$

The area of the 4 sides is $4sh$

You want to minimize $M=s^2+4sh$

The volume must be 32000 so $s^2h=32000$

$h=\frac{32000}{s^2}$

$M=s^2+\frac{128000}{s}$

Find the derivative of M, set it equal to 0 and solve for s. Then solve for h.