Question

what is the volume of the composite figure? express the answer in terms of π. 144π mm³ 168π mm³ 312π mm³ 456π mm³

Explanation:

Step1: Find volume of the cone

The formula for the volume of a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h$. The radius $r = 6$ mm and the height of the cone $h=14 - 6=8$ mm. So $V_{cone}=\frac{1}{3}\pi\times6^{2}\times8=\frac{1}{3}\pi\times36\times8 = 96\pi$ $mm^{3}$.

Step2: Find volume of the hemisphere

The formula for the volume of a hemisphere is $V_{hemisphere}=\frac{2}{3}\pi r^{3}$. With $r = 6$ mm, $V_{hemisphere}=\frac{2}{3}\pi\times6^{3}=\frac{2}{3}\pi\times216 = 144\pi$ $mm^{3}$.

Step3: Find volume of the composite - figure

The volume of the composite figure $V = V_{cone}+V_{hemisphere}$. So $V=96\pi+144\pi = 240\pi$ $mm^{3}$. However, there seems to be an error in the provided options. If we assume the height of the cone is 14 mm (from the top - most point to the center of the base of the hemisphere), then:

New Step1: Find volume of the cone

$V_{cone}=\frac{1}{3}\pi r^{2}h$, with $r = 6$ mm and $h = 14$ mm. So $V_{cone}=\frac{1}{3}\pi\times6^{2}\times14=\frac{1}{3}\pi\times36\times14=168\pi$ $mm^{3}$.

New Step2: Find volume of the hemisphere

$V_{hemisphere}=\frac{2}{3}\pi r^{3}$, with $r = 6$ mm. So $V_{hemisphere}=\frac{2}{3}\pi\times6^{3}=144\pi$ $mm^{3}$.

New Step3: Find volume of the composite - figure

$V = V_{cone}+V_{hemisphere}=168\pi+144\pi=312\pi$ $mm^{3}$.

Answer:

$312\pi$ $mm^{3}$ (corresponding to the third option)