To find the volume of a sphere with a radius of 6 units, we use the formula for the volume of a sphere, which is given by:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere.
Let's evaluate the given expressions to see which one matches this formula.
1. [tex]\(\frac{3}{4} \pi (6)^2\)[/tex]:
[tex]\[
\frac{3}{4} \pi (6)^2 = \frac{3}{4} \pi (36) = 27 \pi
\][/tex]
Numerically:
[tex]\[
\approx 84.823
\][/tex]
2. [tex]\(\frac{4}{3} \pi (6)^3\)[/tex]:
[tex]\[
\frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = \frac{864 \pi}{3} = 288 \pi
\][/tex]
Numerically:
[tex]\[
\approx 904.779
\][/tex]
3. [tex]\(\frac{3}{4} \pi (12)^2\)[/tex]:
[tex]\[
\frac{3}{4} \pi (12)^2 = \frac{3}{4} \pi (144) = 108 \pi
\][/tex]
Numerically:
[tex]\[
\approx 339.292
\][/tex]
4. [tex]\(\frac{4}{3} \pi (12)^3\)[/tex]:
[tex]\[
\frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = \frac{6912 \pi}{3} = 2304 \pi
\][/tex]
Numerically:
[tex]\[
\approx 7238.229
\][/tex]
Finally, we compare the calculated volume of the sphere [tex]\( \frac{4}{3} \pi (6)^3 \approx 904.779 \)[/tex] to each of the options above.
Therefore, the expression that correctly represents the volume of the sphere with radius 6 units is:
[tex]\[
\boxed{\frac{4}{3} \pi(6)^3}
\][/tex]
