# Question #2a36d

Oct 2, 2016

approximately $98.53 \textcolor{w h i t e}{i} c {m}^{3}$

#### Explanation:

Step 1
Imagine a cube box.

Based on this image, we can see that there are $6$ sides.

To find the maximum possible volume for the box, we must determine two things - the area of one side and the side length. Knowing that there is $128 \textcolor{w h i t e}{i} c {m}^{2}$ of material, we can divide $128 \textcolor{w h i t e}{i} c {m}^{2}$ by $6$ to give us the area for one side of the box.

$128 \textcolor{w h i t e}{i} c {m}^{2} \div 6$
$= \frac{64}{3} \textcolor{w h i t e}{i} c {m}^{2}$

Step 2
We can now use the calculated value to determine the side length of the box. Recall that the area of a square is given by:

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} A = {s}^{2} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\underline{\text{where:}}$
$A =$area
$s =$side length

Now that we have determined a relationship between area and side length, we can plug our values into the formula to determine the side length.

$A = {s}^{2}$

$s = \pm \sqrt{A}$

$s = \pm \sqrt{\frac{64}{3}}$

Note: Since a measurement cannot be negative, the only valid answer is $\frac{8}{\sqrt{3}}$ and not $- \frac{8}{\sqrt{3}}$ !

$s = \frac{8}{\sqrt{3}}$

Step 3
Since we have now determined the side length, we can use the formula for volume of a cube to determine its maximum possible volume:

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} V = {s}^{3} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\underline{\text{where:}}$
$V =$volume
$s =$side length

Plugging in the values,

$V = {s}^{3}$

$V = {\left(\frac{8}{\sqrt{3}}\right)}^{3}$

$V = \frac{512}{3 \sqrt{3}}$

$V \approx \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{98.53 \textcolor{w h i t e}{i} c {m}^{3}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$