Question

which formula can be used to find the volume, v, of the composite solid figure represented in the drawing? a. $v = \frac{4}{3}\pi r^3 + \frac{1}{2}\pi r^2 h$ b. $v = \frac{2}{3}\pi r^3 + \frac{1}{2}\pi r^2 h$ c. $v = \frac{4}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$ d. $v = \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$

Explanation:

Step1: Identify the composite solid

The figure is a composite of a hemisphere (half of a sphere) and a cone.

Step2: Volume of a hemisphere

The volume of a full sphere is $V_{sphere} = \frac{4}{3}\pi r^3$. A hemisphere is half of a sphere, so its volume is $V_{hemisphere} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$.

Step3: Volume of a cone

The volume of a cone is given by $V_{cone} = \frac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height.

Step4: Volume of the composite solid

To find the volume of the composite solid, we add the volume of the hemisphere and the volume of the cone. So, $V = V_{hemisphere} + V_{cone} = \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$.

Answer:

D. $V = \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$