Question

The sphere in the picture below is fit into a cube. Find the volume of the sphere and the cube then tell how much of the volume is in the space between the sphere and the cube. Round your volumes to the nearest whole number. USE 3.14 for pi.

Hint: The volume of the cube is equal to the length of theside/edge cubed.

Volume of a cube=s3
Given
6cm
6cm
6cm
r?

Answer 1

To find the volumes of the sphere and the cube and then determine the volume of the space between them, we can follow these steps:

### Step 1: Calculate the Volume of the Cube

The side length of the cube (s) is given as 6 cm. The volume \( V_{cube} \) of a cube is calculated as:

\[
V_{cube} = s^3 = 6^3 = 216 \, \text{cm}^3
\]

### Step 2: Calculate the Volume of the Sphere

For the sphere, the radius (r) is half the side length of the cube, since the sphere fits perfectly inside the cube. Therefore, the radius is:

\[
r = \frac{s}{2} = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm}
\]

The volume \( V_{sphere} \) of a sphere is calculated using the formula:

\[
V_{sphere} = \frac{4}{3} \pi r^3
\]

Substituting \( r = 3 \) cm and \( \pi = 3.14 \):

\[
V_{sphere} = \frac{4}{3} \times 3.14 \times (3^3) = \frac{4}{3} \times 3.14 \times 27
\]

Calculating \( 3^3 \):

\[
3^3 = 27
\]

Now plug it in:

\[
V_{sphere} = \frac{4}{3} \times 3.14 \times 27 \approx \frac{4 \times 3.14 \times 27}{3}
\]

Calculating \( 4 \times 3.14 \):

\[
4 \times 3.14 = 12.56
\]

Continuing with the calculation:

\[
V_{sphere} \approx \frac{12.56 \times 27}{3} = \frac{339.12}{3} \approx 113.04 \, \text{cm}^3
\]

Rounding to the nearest whole number:

\[
V_{sphere} \approx 113 \, \text{cm}^3
\]

### Step 3: Calculate the Volume of the Space Between the Sphere and the Cube

The volume of the space between the sphere and cube can be found by subtracting the volume of the sphere from the volume of the cube:

\[
V_{space} = V_{cube} - V_{sphere}
\]

Substituting the volumes we calculated:

\[
V_{space} = 216 \, \text{cm}^3 - 113 \, \text{cm}^3 = 103 \, \text{cm}^3
\]

### Final Results

- Volume of the cube: \( \approx 216 \, \text{cm}^3 \)
- Volume of the sphere: \( \approx 113 \, \text{cm}^3 \)
- Volume of the space between the sphere and the cube: \( \approx 103 \, \text{cm}^3 \)

Thus, the final answers are:

- Volume of the cube: **216 cm³**
- Volume of the sphere: **113 cm³**
- Volume of the space between the sphere and the cube: **103 cm³**