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:

To find the length of the diagonal, we need the side length.

I) Gives us the volume of the cube. Take the cubed root to find the side length.

II) Gives us the surface area, divide by 6 (the number of faces in a cube) and take the square root to find the side length.

Diagonal of a cube is the side length times the square root of three. Alternatively, this can be found using the Pythagorean Theorem twice.

Either way, we can use I) or II) to find our side lenght and then our diagonal.

The diagonal of a rectangular prism is found via the formula 

The second statement reduces this to 

However, the actual length is unknown. Statement 1 allows the calculation of a numerical value:

Volume of a prism is found by:

We are given the volume in statement I.

We are told how the sides relate in statement II.

Put together these two statements will allow us to set up an equation to find the middle side.

A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of . Find the length of the longest side given the following:

I) The shortest side will be  the length of the medium side

II) The middle side will be 2 feet long

Use I) and II) to find the length of the smallest side

Next, use the short and medium sides, along with info in the prompt, to find the last side: