In order to solve this problem, we must know how to find the diagonal of a cube.

There is an equation to memorize for the diagonal of a cube (this works ONLY for cubes). $\(d=s \sqrt{3}\)$ where $d=$ diagonal and $s=$ length of the side.

Since we know that the side $=7.5$, the diagonal is $\(7.5 \sqrt{3}\)$, so anything greater than or equal to this will be our correct answer.

However, if we don't know this formula, we can figure the answer out via the Pythagorean theorem.

If we draw a diagram of a cube, we can see that the diagonal is actually the hypotenuse of a right triangle whose legs are (1) the diagonal of an entire face of the cube and (2) the side of the cube.

First, we must find the diagonal of one face of the cube. Let $c=$ diagonal of a face of the cube.

$$
\(\begin{aligned}
7.5^2+7.5^2 & =c^2 \\
56.25+56.25 & =c^2 \\
112.5 & =c^2
\end{aligned}\)
$$


So $\(c=\sqrt{112.5}\)$
Now that we have the diagonal of the face, we can figure out the diagonal of the entire cube. The sides of the right triangle are $c$ and $s$; the hypotenuse is the diagonal of the cube.
So let $d=$ diagonal of the cube.

$$
\(\begin{aligned}
c^2+s^2 & =d^2 \\
(\sqrt{112.5})^2+(7.5)^2 & =d^2 \\
112.5+56.25 & =168.75=d^2
\end{aligned}\)
$$


We can either reduce this, or just figure out the numerical value of the square root of 168.75.

It ends up being $\(7.5 \sqrt{3}\)$, or almost 13 .