The total area of a cube is expressed by A(x) = 24x^2+24x+6. What is the volume of this cube?

Mar 2, 2018

$8 {x}^{3} + 12 {x}^{2} + 6 x + 1$

Explanation:

I'm going to assume you meant the surface area is given by $A \left(x\right)$.

We have $A \left(x\right) = 24 {x}^{2} + 24 x + 6$

The formula for the surface area of a cube is given by $6 {k}^{2}$, where $k$ is the length of a side.

We can say that:

$6 {k}^{2} = 24 {x}^{2} + 24 x + 6$

${k}^{2} = 4 {x}^{2} + 4 x + 1$

${k}^{2} = {\left(2 x + 1\right)}^{2}$

$k = 2 x + 1$

So the length of a side is $2 x + 1$.

On the other hand, $V \left(x\right)$, the volume of he cube, is given by ${k}^{3}$.

Here, $k = 2 x + 1$

So we can say:

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

$V \left(x\right) = {\left(2 x + 1\right)}^{2} \left(2 x + 1\right)$

$V \left(x\right) = \left(2 x + 1\right) \left(4 {x}^{2} + 4 x + 1\right)$

$V \left(x\right) = 8 {x}^{3} + 12 {x}^{2} + 6 x + 1$

So the volume of this cube is given by $8 {x}^{3} + 12 {x}^{2} + 6 x + 1$