If we let \(\displaystyle x\) represent the length of an edge on the smaller cube, its volume is \(\displaystyle x^{3}\).

The larger cube has edges three times as long, so the length can be represented as \(\displaystyle 3x\). The volume is \(\displaystyle (3x)^{3}\), which is \(\displaystyle 27x^{3}\).

The large cube's volume of \(\displaystyle 27x^{3}\) is 27 times as large as the small cube's volume of \(\displaystyle x^{3}\).

The volume of a cube is

\(\displaystyle V = s^3\)

So with a side length of 7 inches, the volume is

\(\displaystyle V = 7^{3}\, in^3 = 343\, in^3\)

The volume of a cube is

\(\displaystyle V = s^3\)

So with a side length of 2 ft, the volume is

\(\displaystyle V = 2^{3}\, ft^3 = 8\, ft^3\)

The volume of a cube can be determined through the equation \(\displaystyle V = s^3\), where \(\displaystyle s\) stands for the length of one side of the cube. The equation is \(\displaystyle s\cdot s\cdot s\) because all edges in a cube are the same length. The value for the given edge just needs to be substituted into the equation for \(\displaystyle s\) in order to solve for the cube's volume.

\(\displaystyle V = s^3\)

\(\displaystyle V = 7^3\)

\(\displaystyle V = 7 \cdot 7\cdot 7\)

\(\displaystyle V = 343\:in^3\)

To find the volume of a cube we just multiply a length of one of the cube's sides by itself three times, or in other words, cube it. If we call the length of a side of a cube \(\displaystyle a\), then:

\(\displaystyle Volume=a^3\)

The side of our cube is \(\displaystyle 1\:m\) in length, so we can substitute this into the equation and solve:

\(\displaystyle Volume=a^3=(1\:m)^3=1\:m^3\)

This is a great problem becaue ther are two ways to approach it. The simplest way is to realize that because each edge of the Rubik's Cube is made up of 3 of the smaller cubes we can find the edge length for the whole Rubik's Cube:

\(\displaystyle 1.5 cm + 1.5 cm + 1.5 cm = 4.5 cm\) (edge of whole Rubik's Cube)

Now that we know the edge length finding the volume is easy, we simply multiply the length, width, and height of the cube to find the volume.  This is easy because the length, width, and height are all 4.5 cm.

\(\displaystyle 4.5 \; cm \times 4.5 \; cm \times 4.5 \; cm = 91.125 \; cm^3\) (vol. of Rubik's Cube)

This is the answer.  Another way to approach the problem would be by finding the volume of one of the smaller cubes, then multiplying by 9 to find the volume of one layer, then multiplying by three, because there are 3 layers.

\(\displaystyle 1.5 \; cm \times 1.5 \; cm \times 1.5 \; cm = 3.375\; cm^3\) (vol. of one small cube)

\(\displaystyle 3.375 \; cm^3 \times 9 = 30.375 \;cm^3\) (vol. of one layer of Rubik's Cube)

\(\displaystyle 30.375 \; cm^3 \times 3 = 91.125 \; cm^3\) (vol. of Rubik's Cube)

Two different methods and we got the same answer!

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

\(\displaystyle \text{Volume of Cube}=\text{side}^3\)

Plug in the given side length of the cube to find the volume.

\(\displaystyle \text{Volume of Cube}=8^3=512\)

Next, recall how to find the volume of a cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}\)

Use the given diameter to find the length of the radius.

\(\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}=\frac{6}{2}=3\)

Find the volume of the cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times 3^2 \times 8=72\pi\)

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

\(\displaystyle \text{Volume of Figure}=512-72\pi=285.81\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

\(\displaystyle \text{Volume of Cube}=\text{side}^3\)

Plug in the given side length of the cube to find the volume.

\(\displaystyle \text{Volume of Cube}=10^3=1000\)

Next, recall how to find the volume of a cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}\)

Use the given diameter to find the length of the radius.

\(\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}=\frac{5}{2}=2.5\)

Find the volume of the cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times 2.5^2 \times 10=62.5\pi\)

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

\(\displaystyle \text{Volume of Figure}=1000-62.5\pi=803.65\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

\(\displaystyle \text{Volume of Cube}=\text{side}^3\)

Plug in the given side length of the cube to find the volume.

\(\displaystyle \text{Volume of Cube}=12^3=1728\)

Next, recall how to find the volume of a cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}\)

Use the given diameter to find the length of the radius.

\(\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}=\frac{8}{2}=4\)

Find the volume of the cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times 4^2 \times 12=192\pi\)

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

\(\displaystyle \text{Volume of Figure}=1728-192\pi=1124.81\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

\(\displaystyle \text{Volume of Cube}=\text{side}^3\)

Plug in the given side length of the cube to find the volume.

\(\displaystyle \text{Volume of Cube}=5^3=125\)

Next, recall how to find the volume of a cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}\)

Use the given diameter to find the length of the radius.

\(\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}=\frac{3}{2}=1.5\)

Find the volume of the cylinder.

\(\displaystyle \text{Volume of Cylinder}=\pi\times 1.5^2 \times 5=11.25\pi\)

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

\(\displaystyle \text{Volume of Figure}=125-11.25\pi=89.66\)

Make sure to round to \(\displaystyle 2\) places after the decimal.