To find the volume of a sphere, we use the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the sphere.
Given that the diameter of the sphere is 12 ft, we first need to find the radius. The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{\text{diameter}}{2} \][/tex]
[tex]\[ r = \frac{12 \text{ ft}}{2} \][/tex]
[tex]\[ r = 6 \text{ ft} \][/tex]
Now we substitute the radius back into the volume formula:
[tex]\[ V = \frac{4}{3} \pi (6 \text{ ft})^3 \][/tex]
Next, we need to calculate [tex]\( (6 \text{ ft})^3 \)[/tex]:
[tex]\[ (6 \text{ ft})^3 = 6 \text{ ft} \times 6 \text{ ft} \times 6 \text{ ft} = 216 \text{ ft}^3 \][/tex]
Now we substitute [tex]\( 216 \text{ ft}^3 \)[/tex] into the volume formula:
[tex]\[ V = \frac{4}{3} \pi (216 \text{ ft}^3) \][/tex]
Simplifying further:
[tex]\[ V = \frac{4}{3} \pi \times 216 \text{ ft}^3 \][/tex]
[tex]\[ V = \frac{4 \times 216 \pi}{3} \text{ ft}^3 \][/tex]
[tex]\[ V = \frac{864 \pi}{3} \text{ ft}^3 \][/tex]
[tex]\[ V = 288 \pi \text{ ft}^3 \][/tex]
Thus, the exact volume of the sphere is:
[tex]\[ 288 \pi \text{ ft}^3 \][/tex]
