# Thread: Prove surface area of cuboid.

1. ## Prove surface area of cuboid.

A cuboid has a volume of $\displaystyle 8 m^3$. The base of the cuboid is square with sides of length x metres. the sureface area of the cuboid is $\displaystyle A m^2$.
Show that $\displaystyle A=2x^2+\frac{32}{x}$
thanks!

2. Originally Posted by BabyMilo
thanks!
The side lengths are $\displaystyle x, x, y$.

The volume is

$\displaystyle V = x^2y$

So $\displaystyle 8 = x^2y$

$\displaystyle y = \frac{8}{x^2}$.

The surface area is

$\displaystyle TSA = 2x^2 + 4xy$

So $\displaystyle A = 2x^2 + 4xy$

$\displaystyle A = 2x^2 + 4x\left(\frac{8}{x^2}\right)$

$\displaystyle A = 2x^2 + \frac{32}{x}$.

3. Originally Posted by BabyMilo
thanks!

What is known:

Code:
$\displaystyle V = 8m^3$
square base.

$\displaystyle A_s = Am^2$
Use the formula for the volume of a cube, given that you have a square base an $\displaystyle x^2$ term should appear. It will be easiest to introduce a new variable for the height - $\displaystyle h$ in my case.

Spoiler:
$\displaystyle V = x^2h = 8 \: \: \rightarrow \: \: h = \frac{8}{x^2}$ where $\displaystyle h$ is the height of the cuboid

You can now get an expression for the surface area given that it is equal to the sum of it's faces. If you draw a diagram you should be able to find the expression more easily.

Spoiler:
$\displaystyle A = 2x^2 + 4xh$

You have simultaneous equations to solve now. Eliminate h as it's not being asked for and get the expression for A

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# derive lateral surface area of cuboid

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