Which equation correctly expresses the volume of a hemisphere?

The volume of a hemisphere is derived from the formula for the volume of a complete sphere, which is given by ( V = \frac{4}{3} \pi r^3 ). Since a hemisphere is exactly half of a sphere, the correct expression for the volume of a hemisphere must represent half of the sphere's volume.

To find the volume of a hemisphere, you take the volume of the sphere and divide it by 2:

[

V_{hemisphere} = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3

]

This simplifies to the expression ( V = \frac{2 \pi r^3}{3} ), which corresponds to the equation provided. Therefore, this equation accurately describes the volume of a hemisphere based on the relationship to the sphere.

Other choices do not correctly represent the volume of a hemisphere. The first choice gives the volume of a full sphere, while the second one incorrectly suggests that simply dividing the sphere’s volume by two should be expressed differently. The fourth option also misrepresents the dimensions involved, adding confusion with an incorrect exponent. Thus,