Call the sidelength, surface area, and volume of the cube \(\displaystyle s\), \(\displaystyle A\), and \(\displaystyle V\), respectively.

Then 

\(\displaystyle V = s^{3}\)

or, equivalently, 

\(\displaystyle s = \sqrt[3]{V}\)

So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be

\(\displaystyle s = \sqrt[3]{1,728} = 12\)

Also,

\(\displaystyle A = 6s^{2}\)

or, equivalently,

\(\displaystyle s = \sqrt{\frac{A}{6}}\)

Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be

 \(\displaystyle s = \sqrt{\frac{864}{6}} = \sqrt{144} = 12\)

Therefore, the answer is that either statement alone is sufficient.

The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula \(\displaystyle V = \frac{4}{3} \pi r^{3}\) can be used to find the volume of the sphere.

In order to find the length of an edge, we would need any information about one of the faces of the cube or about the diagonal of the cube.

Statement 1 gives us the length of the diagonal of the cube, since the formula for the diagonal is \(\displaystyle d=s\sqrt{3}\) where \(\displaystyle s\) is the length of an edge of the cube and \(\displaystyle d\) is the length of the diagonal we are able to find the length of the edge. Therefore statement 1 alone is sufficient.

Statement 2 alones is insufficient, it gives us something we can already tell knowing that ABCDEFGH is a cube.

Statement 1 alone is sufficient.

Like we have previously seen, to find the length of an edge, we need to have information about the other faces or anything else within the cube.

Statement 1 tells us that the volume of the cube is \(\displaystyle 27\), from this we can find the length of the side of the cube. Statement 1 alone is sufficient.

Statement 2, tells us that the area of ABCD is \(\displaystyle 9\), similarily, by taking the square root of this number, we can find the length of the edge of the cube.

Therefore each statement alone is sufficient.

Statement 1:  Use the volume formula for a cube to solve for the side length.

      \(\displaystyle V=a^3=125\)  where \(\displaystyle a\) represents the length of the edge

      \(\displaystyle a=\sqrt[3]{125}=5cm\)

Statement 2:   Use the surface area formula for a cube to solve for the side length.

   \(\displaystyle SA=6a^2=150\)

   \(\displaystyle a^2=25\)

   \(\displaystyle a=5cm\)

Each statement alone is sufficient to answer the question.

To know the diagonal HB of the cube, we need to have information about the length of the edges of the cube. Knowing the area of a face would allow us to find the length of an edge; therefore statement 1 alone is sufficient.

Statment 2 alone is sufficient as well since the length of a diagonal of a square is given by \(\displaystyle s\sqrt{2}\), where \(\displaystyle s\) is the length of a side of the square and ultimately we can find the length of an edge of the square.

Therefore each statement alone is sufficient.

Statement 1: We can use the surface area to find the length of the cube's edge and then determine the length of the diagonal.

              \(\displaystyle SA=6a^2=96\)

where \(\displaystyle a\) represents the length of the cube's edge

\(\displaystyle a^2=16\)

\(\displaystyle a=4\)

Now that we know the length of the edge, we can find the length of the diagonal: \(\displaystyle d=a\sqrt{3}=4\sqrt{3}cm\).

Statement 2: Once again, we can use the provided information to find the length of the cube's edge.

             \(\displaystyle V=a^3=64\)

             \(\displaystyle a=\sqrt[3]{64}=4\)

Which means the length of the diagonal is \(\displaystyle d=a\sqrt{3}=4\sqrt{3} cm\).

Thus, each statement alone is sufficient to answer the question.

Statement 1: To find the length of the diagonal of the cube we need to find the length of the cube's edge. We can use the given surface area value to do so:

\(\displaystyle SA=6a^2=294\)

\(\displaystyle a^2=49\)

\(\displaystyle a=7\)

Now that we know the length of the cube's edge, we can calculate the diagonal:

\(\displaystyle d=a\sqrt{3}=7\sqrt{3}\approx 12.12cm\)

Statement 2: We're given the volume of the cube which we can also use to solve for the length of the cube's edge.

\(\displaystyle a^3=343\)

\(\displaystyle a=\sqrt[3]{343}=7\)

Knowing the length of the cube's edge allows us to calculate the diagonal:

\(\displaystyle d=a\sqrt{3}=7\sqrt{3}\approx 12.12cm\)

Each statement alone is sufficient to answer the question.

Statement 1: We need the length of an edge in order to find the diagonal of the cube. Luckily, we can find the length using the information provided:

\(\displaystyle SA=6a^2=726\)

\(\displaystyle a^2=121\)

\(\displaystyle a=11\)

Now that we know the length of an edge is 11 cm we can find the length of the diagonal.

\(\displaystyle d=a\sqrt{3}=11\sqrt{3}cm\)

Statement 2: We're given the information we need to find the diagonal, all we need to do is plug it into the equation

\(\displaystyle d=a\sqrt{3}=11\sqrt{3}cm\)

Statement 1: The information provided in the question is only useful if we're given a relationship between cube A and cube B. Since this statement does provide us with the ratio of 1:2, we can answer the question.

\(\displaystyle d=a\sqrt{3}=14\sqrt{3}\)

where \(\displaystyle a\) represents the length of the cube's edge

we can easily see the length measures \(\displaystyle 14cm\)

Remember the ratio of cube A to cube B is 1:2.

\(\displaystyle \frac{1}{2}=\frac{a}{14}\)

\(\displaystyle 2a=14\)

\(\displaystyle a=7cm\)

Now that we know the length, we can find the diagonal of the cube:

\(\displaystyle d=a\sqrt{3}=7\sqrt{3}cm\)

Statement 2: We're given information about cube A so we don't need to worry about cube B. Using this information we can solve for the edge length of cube A and then calculate the diagonal.

\(\displaystyle SA=6a^2=294\)

\(\displaystyle a^2=49\)

\(\displaystyle a=7cm\)

Knowing the length of the edge allows us to find the diagonal of the cube

\(\displaystyle d=a\sqrt{3}=7\sqrt{3}cm\)