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