Question

If 30,000 cm2 of material is available to make a box with a square base and an open top, what is the largest possible volume (in cm3) of the box?

Answer 1

The largest possible volume of the box is 2000000 cubic meters.

Explanation:

The volume (
`V`), in cubic centimeters, and surface area (
`A_(s)`), in square centimeters, of the box with a square base are described below:

`A_(s) = l^(2)+h\cdot l` (1)

`V = l^(2)\cdot h` (2)

Where:

`l` - Side length of the base, in centimeters.

`h` - Height of the box, in centimeters.

By (2), we clear
`h` within the formula:

`h = (V)/(l^(2))`

And we apply in (1) and simplify the resulting expression:

`A_(s) = l^(2)+ (V)/(l)`

`A_(s)\cdot l = l^(3)+V`

`V = A_(s)\cdot l -l^(3)` (3)

Then, we find the first and second derivatives of this expression:

`V' = A_(s)-3\cdot l^(2)` (4)

`V'' = -6\cdot l` (5)

If
`V' = 0` and
`A_(s) = 30000\,cm^(2)`, then we find the critical value of the side length of the base is:

`30000-3\cdot l^(2) = 0`

`3\cdot l^(2) = 30000`

`l = 100\,cm`

Then, we evaluate this result in the expression of the second derivative:

`V'' = -600`

By Second Derivative Test, we conclude that critical value leads to an absolute maximum. The maximum possible volume of the box is:

`V = 30000\cdot l - l^(3)`

`V = 2000000\,cm^(3)`

The largest possible volume of the box is 2000000 cubic meters.