The weight of a cube is dependent on its density in pounds per cubic foot and its volume in cubic feet. We need Statement 1 for the density. Statement 2 is needed for the volume - and it gives us the means to find it, since we can take the square root of the area of one side, 16, to get sidelength 4 feet, and we can cube that to get the volume of 64 cubic feet. Now we can multiply 64 cubic feet by 3 pounds per cubic foot to get 192 pounds.

If we know the base of W, we can find the side length. We then cube the side length to find the volume of a square. 

If we know the diagonal of the base of W, we can find the side length. As outlined above, we can then cube the side length to find the volume.

Therefore either statement alone is sufficient to solve the question.

Statement 1: We need the length of the edge of the cube to calculate the volume. In this statement we're provided with the length so we just need to plug it into the equation for a cube's volume.

Statement 2: In this case, we need to solve for the length of the cube's edge which we can easily do: 

Now that we have our length, we can calculate the volume.

Recall that the volume of a cube is equal to the cube of its side length, and that the diagonal of a cube is equal to  length. 

I) Use the following:

II) Gives us the diagonal. Divide by the square root of three, then cube it!

Either statement is sufficient to answer the question.

To know the volume of the box, which is shaped like a rectangular prism, you multiply the length, the width, and the height together. Neither statement alone gives you those dimensions, just clues as to how they are related. But together, you can form a system of linear equations using height  and width (and length) .

By the first and second statements, respectively, 

This system can be solved:

From this, you can determine the length and height, and, subsequently, the volume.

To find surface area, we need to know the side length of the box.

We are told the box has equal length, width and height. This means it is a cube.

I) Gives us the diagonal of the cube, from this we can find the side length.

, where s is the side length.

Now we can use the side length to find the surface area of the box.

II) Gives us the volume of the cube, from which we can also find the side length.

From here we can find the surface area.

Thus, either statement is sufficient.

Fry recently met a robot known for bending things. The robot wants to make a box out of steel by bending a single sheet of metal. Find the total area of the sheet of metal given the following:

I) The box will be  long

II) The box's height will be 2 feet taller than its length, and the box's width will be  less than its length.

This question is a surface area question in disguise. To find the surface area of a prism, we need the area of each side. Begin by using I and II to find each side length:

Length: 36 inches or 3 feet

Height:  or 5 feet

Width:  or 

So, we have all our side lengths. Next, use the following formula for surface area of a rectangular prism:

Where l,w, and h are length, width and height

Given base area  height , the volume of either a pyramid or a cone is equal to

.

If we know the heights of the two figures are the same, we still need to know which has the greater base area in order to know which has the greater volume, so Statement 1 is insufficient. By a similar argument, Statement 2 is insufficient. 

If we know both heights are equal and that both bases have equal area, then we know that the volumes are equal.

Statement 1:  , , 

The height can be easily solved for by substituting the length and width. Height is 2, and the volume can be calculated by the following formula.

Statement 2:  The diagonal is 6.

Write the equation of the diagonal in terms of length, width, and height.

Although the lengths from Statement 1 will yield a diagonal of six, there are multiple combinations of length, width, and height which will also give a diagonal of six.  The product of these three dimensions may or may not give a volume of six.  Statement 2 has insufficient information to calculate the volume.

Therefore:

To find the volume of a rectangular solid, use the following formula:

Statement I relates the length, width and height of the treasure chest.

Statement II gives us the length of the shortest side of the treasure chest.

Use Statement II and the relationships described in Statement I to find each side, then multiply them all together to get your answer.

If the shortest side is 2.5ft, then the longest side must be 5ft. 

If the longest side is 5ft, then the medium side can be found via the following:

So the volume is as follows: