To determine the correct formula for the volume of a sphere, let's examine each option carefully:
(A) [tex]\(\pi r^2 h\)[/tex]
- This formula represents the volume of a cylinder, not a sphere. The volume of a cylinder is calculated with the formula [tex]\(V = \pi r^2 h\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
(B) [tex]\(\frac{2}{3} \pi x^3\)[/tex]
- This formula does not correspond to the volume of a sphere. The coefficient [tex]\(\frac{2}{3}\)[/tex] is incorrect, and the variable [tex]\(x\)[/tex] should be noted specifically as the radius [tex]\(r\)[/tex].
(C) [tex]\(\frac{1}{3} \pi r^2 / 2\)[/tex]
- This expression is incorrect. Simplifying it yields [tex]\(\frac{1}{6} \pi r^2\)[/tex], which is not the formula for the volume of a sphere. This seems more aligned with a formula used for conical shapes than for spheres.
(D) [tex]\(\frac{4}{3} \pi r^3\)[/tex]
- This is the correct formula for the volume of a sphere. The volume [tex]\(V\)[/tex] of a sphere is calculated using [tex]\(V = \frac{4}{3} \pi r^3\)[/tex], where [tex]\(r\)[/tex] is the radius.
Given the mathematical principles for the volume of a sphere, the correct option is:
(D) [tex]\(\frac{4}{3} \pi r^3\)[/tex]
