# How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of 8,788 (cm)^3?

Feb 27, 2015

You set $x$ as being the sides, and $h$ for the height.

The box will have a square bottom.
Then the amount of cardboard used will be:
For the bottom: $x \cdot x = {x}^{2}$
For the sides: $x \cdot h \cdot 4$(sides)$= 4 x h$

Total area : $A = {x}^{2} + 4 x h$

The volume of the box= $x \cdot x \cdot h = 8788$ from which we can conclude that $h = \frac{8788}{x} ^ 2$

Substituting that into the formula for the area $A$, we get:

$A = {x}^{2} + 4 x \cdot \left(\frac{8788}{x} ^ 2\right) = {x}^{2} + \frac{35152}{x}$

To find the minimum, we have to differentiate and set to $0$

$A ' = 2 x - \frac{35152}{x} ^ 2 = 0 \to 2 x = \frac{35152}{x} ^ 2$ multiply by ${x}^{2}$

$2 {x}^{3} = 35152 \to {x}^{3} = 17576 \to x = \sqrt[3]{17576} = 26$
Substitute: $h = 8788 / {26}^{2} = 13$

The sides will be $26 c m$ and the height will be $13 c m$