Since we are looking at inches, we will look at one foot as twelve inches.

Let $\displaystyle s$ be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem, 

$\displaystyle s^{2}+s^{2}+s^{2}=12^{2}$

$\displaystyle 3s^{2}=144$

$\displaystyle s^{2}=48$

$\displaystyle s= \sqrt{48} = 4\sqrt{3}$ inches.

Cube this sidelength to get the volume:

$\displaystyle s^{3}= \left ( 4\sqrt{3} \right )^{3} = 4^{3} \cdot \left ( \sqrt{3} \right ) ^{3} = 64 \cdot 3 \sqrt{3}= 192 \sqrt{3}$ cubic inches.

To find the volume of a cube we use the formula:

$\displaystyle V=s^3$

In our case the total volume of the pool is 10³ cubic meters.

Filling the pool to 95% of its total volume will require: $\displaystyle 0.95\times1000 = 950$ cubic meters.

Now we need to convert from cubic meters to liters in order to answer the question.

Remember $\displaystyle 1$ cubic meter = $\displaystyle 1000$ liters.

Thus,

$\displaystyle 950 \times 1000=950,000$.

Therefore, the number of liters of water needed is 950000 liters.

The diameter of a sphere is equal to the length of an edge of the cube in which it is inscribed. We can derive the radius using the formula for the surface area of a sphere:

$\displaystyle SA = 4 \pi r^{2}$

Substituting in the given value:

$\displaystyle 4 \pi r^{2} = 324 \pi$

Dividing each side by $\displaystyle 4\pi$:

$\displaystyle r^{2} = \frac{324 \pi}{4 \pi }= 81$

Taking the square root of each side:

$\displaystyle r = 9$

The diameter is twice this, or $\displaystyle 18$. This is the length of the edge of the cube, so we can cube it to get the volume:

$\displaystyle V =18 ^{3}= 5,832$

The diameter of a sphere is equal to the length of a diagonal of the cube it circumscribes. We can derive the radius using the formula for the volume of a sphere, but first, we have to solve for the radius using the formula for the surface area of a sphere:

$\displaystyle SA = 4 \pi r^{2}$

$\displaystyle 4 \pi r^{2} = 324 \pi$

$\displaystyle r^{2} = \frac{324 \pi}{4 \pi }= 81$

$\displaystyle r = 9$

The diameter is twice this, or $\displaystyle 18$. This is the length of the diagonal of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$\displaystyle s^{2}+ s^{2}+ s^{2} = 18^{2}$

$\displaystyle 3s^{2} = 324$

$\displaystyle s^{2} =108$

$\displaystyle s = \sqrt{108} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}$

The volume is the cube of this:

$\displaystyle V =(6 \sqrt{3}) ^{3}= 6^{3} \cdot \left (\sqrt{3} \right )^{3} = 216 \cdot 3 \cdot \sqrt{3} = 648 \sqrt{3}$, which is not given as one of the choices.

One way we can solve this problem is by first determining the dimensions of the original cube. By definition, the length, width, and depth of a cube are equal, so if the original volume is 8 cubic feet, then:

$\displaystyle V_o=L_o^3=8\rightarrow L_o=\sqrt[3]{8}=2$

We then double the length of each side of the original cube, so the length of each side of the new cube is:

$\displaystyle L=2L_o=2(2)=4$

Now we can use the dimensions of our new cube to find its volume:

$\displaystyle V=L^3=(4)^3=64$

So if we double the dimensions of the original cube, the resulting cube has a volume that is eight times greater, 64 cubic feet.