The two statements are actually equivalent; if  is its volume,  is the area of its base, and  is its height, then , or . So if we know the first statement, that is, , then , which is the second statement. 

To find the radius, use , or, equivalently, 

The answer is that either statement alone is sufficient to answer the question.

The question is essentially this: if  are the volumes of the barrels, then find  such that 

,

or

Equivalently, 

or

This means that knowing both the ratio of the heights and the ratio of the radii (and subsequently, the widths) is necessary and sufficient. Therefore, you need both statements, and both together are sufficient.

Assume both statements. We show that insufficient information is provided using two cases.

Case 1: Cylinder 1 has height 10 and radius 20. Then Cylinder 2 has height 40 and radius 10.

The volume of Cylinder 1 is 

The volume of Cylinder 2 is 

Case 2: Cylinder has height 40 and radius 10. Then Cylinder 2 has height 20 and radius 40.

The volume of Cylinder 1 is 

The volume of Cylinder 2 is 

In one case, Cylinder 1 had greater volume; in the other, Cylinder 2 did. This makes the two statements insufficient.

(1) does not give us any information about the cans' dimension or the length of the box, so it is not sufficient alone.

(2) does not give us box dimensions or any information about the cans' diameter, so it is not sufficient alone.

Both statement taken together give more information but we still dont have complete dimensions for the box or each can.

Therefore the right answer is E.

Recall the formula for volume of a cylinder:

Statement I gives us V

Statement II gives us h

We can then use both of them to find our "r." We cannot do it without both of them.

We are given the radius; if we know the height, we can use the formula

to calculate the volume of the tank.

The second statement gives us that the tank is 30 feet high. But the first statement gives us the way to find the height by using the lateral area formula.

First we have to convert square yards to square feet by multiplying by 9.

Either way, we now have both radius and height, and we can find the volume:

The answer is that either statement alone is sufficient to answer the question.

Call the sidelength, surface area, and volume of the cube , , and , respectively.

Then 

or, equivalently, 

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

Also,

or, equivalently,

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

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  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  where  is the length of an edge of the cube and  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 , 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 , 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.