To begin, you must solve for the height of the original cylinder. We know:

For our values, we know:

Now, divide both sides by :

So, if we have a new radius of , our volume will be:

Side 1: surface area of the 6 x 7 side is 42

Side 2: surface area of the 7 x 8 side is 56

Side 3: surface area of the 6 x 8 side is 48.

We can add sides 1 and 3 to get 90, so that's not the answer.

We can add sides 1 and 1 to get 84, so that's not the answer.

We can add sides 2 and 3 to get 104, so that's not the answer.

We can add sides 2 and 2 to get 112, so that's not the answer.

This leaves the answer of 92. Any combination of the three sides of the rectangular prism will not give us 92 as the total surface area.

First, find the area of the base of the cyclinder:

 and multiply that by two, since there are two sides with this measurement: . 

Then, you find the width of the rectangular portion (label portion) of the prism by finding the circumference of the cylinder: 

. This is then multiplied by the height of the cylinder to find the area of the rectanuglar portion of the cylinder: .

Finally, add all sides together and round: .

Find the area of the triangular sides first: 

Since there are two sides of this area, we multiply the area by 2: 

Next find the area of the rectangular regions. Two of them have the width of 3 feet and a length of 7 feet, while the last one has a width measurement of  feet and a length of 7 feet. Multiply and add all other sides: 

.

Lastly, add the triangular sides to the rectangular sides and round: 

.

Remember, the formula for the volume of a rectangular prism is width times height times length:

Now, let's solve the word problem for each of these values. We know that . If length is double the width, then the length must be 6 units. If the height is two-thirds the length, then the height must be 4:

Multiply all three values together to solve for the volume:

The volume of the rectangular prism is  units cubed.

The surface area of a rectangular prism with sides , , and  is given as:

. 

Two sides are known; it does not matter how they are designated, but for this problem let  and , with  as the unknown side. This yields equality:

Now that the three dimensions are known, it's possible to calculate the volume:

Recall that the equation for the volume of a cube is:

Since the sides of a cube are merely squares, the surface area equation is just  times the area of one of those squares:

So, for our two quantities:

Quantity A

Use your calculator to estimate this value (since you will not have a square root key). This is .

Quantity B

First divide by :

Therefore, 

Therefore, the two quantities are equal.

The surface area of a cube is made up of  squares. Therefore, the equation is merely  times the area of one of those squares.  Since the sides of a square are equal, this is:

, where  is the length of one side of the square.

For our data, we know:

This means that:

Now, while you will not have a calculator with a square root key, you do know that . (You can always use your calculator to test values like this.) Therefore, we know that . This is the length of one side

There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.

Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.

First, we must ascertain the length of each side.  Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x². This yields the equation:

6x² = 486, which simplifies to: x² = 81; x = 9.

Therefore, each side has a length of 9.  Imagine the cube is centered on the origin.  This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5).  To find the distance between these, we use the three-dimensional distance formula:

d = √((x₁ – x₂)² + (y₁ – y₂)² + (z₁ – z₂)²)

For our data, this will be:

√( (–4.5 – 4.5)² + (–4.5 – 4.5)² + (4.5 + 4.5)²) =

√( (–9)² + (–9)² + (9)²) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)