To determine the volume of a sphere, we need to use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume.
- [tex]\( r \)[/tex] is the radius of the sphere.
- [tex]\( \pi \)[/tex] is a constant approximately equal to 3.14159.
First, we need to find the radius of the sphere. The diameter of the sphere is given as 4 centimeters. The radius is half of the diameter, so:
[tex]\[ r = \frac{d}{2} = \frac{4 \, \text{cm}}{2} = 2 \, \text{cm} \][/tex]
Now, we can substitute the radius back into the volume formula:
[tex]\[ V = \frac{4}{3} \pi (2 \, \text{cm})^3 \][/tex]
Next, we calculate the cube of the radius:
[tex]\[ (2 \, \text{cm})^3 = 2^3 \, \text{cm}^3 = 8 \, \text{cm}^3 \][/tex]
Substituting this back into the volume formula gives us:
[tex]\[ V = \frac{4}{3} \pi \times 8 \, \text{cm}^3 \][/tex]
[tex]\[ V = \frac{32}{3} \pi \, \text{cm}^3 \][/tex]
So, the volume of the sphere is:
[tex]\[ V = \frac{32}{3} \pi \, \text{cm}^3 \][/tex]
Among the given options, the correct one is:
[tex]\[ \mathbf{\frac{64}{3} \pi \, \text{cm}^3} \][/tex]
