An open box is to be made out of a piece of a square card board of sides $18 c m$ by cutting off equal squares from the corners and turning up the sides. Find the maximum volume of the box.

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Solution

Let each side of the square cut off from each corner be $x c m$
Then the base of the box will be of side $18 - 2 x c m$ and the height of the box will be $x c m$
Then volume of box $V = (18 - 2 x) (18 - 2 x) x$
$V = {(18 - 2 x)}^{2} x$
$V = 4 x^{3} + 324 x - 72 x^{2}$ ...(i)
Differentiating w.r t to $x$ , we get
$\frac{d V}{d x} = 12 x^{2} + 324 - 144 x$
$\frac{d V}{d x} = 12 (x^{2} - 12 x + 27)$ ....(ii)
For maximum volume $\frac{d V}{d x} = 0$
$\Rightarrow 12 (x^{2} - 12 x + 27) = 0$
$\Rightarrow$ $x^{2} - 9 x - 3 x + 27 = 0$
$\Rightarrow$ $(x - 9) (x - 3) = 0$
$\Rightarrow$ $x = 9, 3$
Again differentiating, we get
$\frac{d^{2} V}{d x^{2}} = 2 x - 12$ ...(iii)
At $x = 9$ ,
$\frac{d^{2} V}{d x^{2}} = + v e$
$∴$ $V$ is minimum at $x = 9$ at $x = 3$
$\frac{d^{2} V}{d x^{2}} = - v e$
$∴$ $V$ is maximum at $x = 3$
$∴$ Maximum volume $V = (18 - 6) (18 - 6) \times 3$
$= 12 \times 12 \times 3 = 432 {c m}^{3}$

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An open box is to be made out of a piece of a square card board of sides 18 cm by cutting off equal squares from the corners and turning up the sides. Find the maximum volume of the box.

An open box is to be made out of a piece of a square card board of sides $18 c m$ by cutting off equal squares from the corners and turning up the sides. Find the maximum volume of the box.