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A box from a sheet

If you cut equal squares from the corners of a square sheet and fold up, the height equals the side of the square. The base is the sheet minus two such sides in each direction. The volume is height times base area, and the domain stops before the base disappears. The derivative has two zeros: one gives zero volume, and the other gives the largest volume.

V(x) = x(s - 2x)^2

the volume

0 < x < s/2

the domain

Symbols

$s$	the side of the sheet	cm
$x$	the side of the corner square	cm
$V$	the volume	cm^3

Example

The sheet is $30$ cm. $V(x) = x(30 - 2x)^2$ , and $V'(x) = 0$ gives $x = 5$ cm. Then the volume is $5 \cdot 20^2 = 2000$ cm $^3$ .

The solution

x = 15

lies at the edge and gives a base of zero, so the volume is zero.

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Part of Mathematics R1: Optimisation.