To determine the volume of a sphere with a radius of 6 units, we use the formula for the volume of a sphere. The volume [tex]\( V \)[/tex] of a sphere with radius [tex]\( r \)[/tex] is given by:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Given that the radius [tex]\( r \)[/tex] is 6 units, we substitute this value into the formula:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
This expression represents the volume of the sphere. To confirm, let's substitute the radius into the formula and compute the volume:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
The term [tex]\( (6)^3 \)[/tex], or 6 cubed, equals 216.
So, the volume equation becomes:
[tex]\[ V = \frac{4}{3} \pi \cdot 216 \][/tex]
Simplifying, we multiply [tex]\( \frac{4}{3} \)[/tex] by 216 to get the coefficient in front of [tex]\( \pi \)[/tex]:
[tex]\[ \frac{4}{3} \times 216 = 288 \][/tex]
Thus, the volume in terms of [tex]\( \pi \)[/tex] is:
[tex]\[ V = 288 \pi \][/tex]
Comparing this expression with the given options:
- [tex]\(\frac{3}{4} \pi (6)^2 \)[/tex]
- [tex]\(\frac{4}{3} \pi (6)^3 \)[/tex]
- [tex]\(\frac{3}{4} \pi (12)^2 \)[/tex]
- [tex]\(\frac{4}{3} \pi (12)^3 \)[/tex]
We see that the correct expression for the volume of a sphere with radius 6 units is:
[tex]\[ \frac{4}{3} \pi (6)^3 \][/tex]
