# How do you find the volume of the box given the dimensions are: 3x + 1, 2x - 1, x + 2?

Aug 20, 2017

See a solution process below:

#### Explanation:

The formula for the volume of a box is:

$V = l \times w \times h$

Where:

$V$ is the volume

$l$ is the length of the box

$w$ is the width of the box

$h$ is the height of the box

We can substitute to give:

$V = \left(\textcolor{red}{3 x} + \textcolor{red}{1}\right) \left(\textcolor{b l u e}{2 x} - \textcolor{b l u e}{1}\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

Expanding gives:

$V = \left(\left(\textcolor{red}{3 x} \times \textcolor{b l u e}{2 x}\right) - \left(\textcolor{red}{3 x} \times \textcolor{b l u e}{1}\right) + \left(\textcolor{red}{1} \times \textcolor{b l u e}{2 x}\right) - \left(\textcolor{red}{1} \times \textcolor{b l u e}{1}\right)\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

$V = \left(6 {x}^{2} - 3 x + 2 x - 1\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

$V = \left(6 {x}^{2} + \left(- 3 + 2\right) x - 1\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

$V = \left(6 {x}^{2} + \left(- 1\right) x - 1\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

$V = \left(6 {x}^{2} - 1 x - 1\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

$V = \left(6 {x}^{2} - x - 1\right) \left(\textcolor{g r e e n}{x} + \textcolor{g r e e n}{2}\right)$

Expanding again gives:

$V = \left(6 {x}^{2} \times \textcolor{g r e e n}{x}\right) + \left(6 {x}^{2} \times \textcolor{g r e e n}{2}\right) - \left(x \times \textcolor{g r e e n}{x}\right) - \left(x \times \textcolor{g r e e n}{2}\right) - \left(1 \times \textcolor{g r e e n}{x}\right) - \left(1 \times \textcolor{g r e e n}{2}\right)$

$V = 6 {x}^{3} + 12 {x}^{2} - {x}^{2} - 2 x - x - 2$

$V = 6 {x}^{3} + 12 {x}^{2} - 1 {x}^{2} - 2 x - 1 x - 2$

$V = 6 {x}^{3} + \left(12 - 1\right) {x}^{2} + \left(- 2 - 1\right) x - 2$

$V = 6 {x}^{3} + 11 {x}^{2} + \left(- 3\right) x - 2$

$V = 6 {x}^{3} + 11 {x}^{2} - 3 x - 2$