# How do you find the volume of the largest open box that can be made from a piece of cardboard 35 inches by 19 inches by cutting equal squares from the corners and turning up the sides?

Apr 5, 2018

Volume of largest open box is $2167 i {n}^{3}$

#### Explanation:

If we make a box from a piece of cardboard $35$ inches by $19$ inches by cutting equal squares from the corners and turning up the sides (ssee g=figure below),

we get a box, whose height is $x$ and length and bradth are $\left(35 - 2 x\right)$ and $\left(19 - 2 x\right)$ and its volume $V$ is given by

$V = x \left(35 - x\right) \left(19 - x\right) = x \left(665 - 54 x + {x}^{2}\right) = - {x}^{3} - 54 {x}^{2} + 665 x$

Volume maximizes when $\frac{\mathrm{dV}}{\mathrm{dx}} = 0$

as $\frac{\mathrm{dV}}{\mathrm{dx}} = - 3 {x}^{2} - 108 x + 665 = 0$

or $3 {x}^{2} + 108 x - 665 = 0$

i.e. $x = \frac{- 108 \pm \left({108}^{2} - 4 \cdot 3 \cdot \left(- 665\right)\right)}{6}$

= $\frac{- 108 \pm \sqrt{19644}}{6} = \frac{- 108 \pm 140.157}{6}$

and omitting negative values $x = \frac{- 108 + 140.157}{6} = \frac{32.157}{6} = 5.36$ inches

and when $x = 5.36$ volume is

$V = 5.36 \left(35 - 5.36\right) \left(19 - 5.36\right) = 5.36 \cdot 29.64 \cdot 13.64 = 2167 i {n}^{3}$