Question

what is the volume of the composite figure? express the answer in terms of \\(\pi\\). \\(\bigcirc\\) \\(144\pi\\ \text{mm}^3\\) \\(\bigcirc\\) \\(168\pi\\ \text{mm}^3\\) \\(\bigcirc\\) \\(312\pi\\ \text{mm}^3\\) \\(\bigcirc\\) \\(456\pi\\ \text{mm}^3\\)

Explanation:

Step1: Identify figure components

The composite figure is a cone + a hemisphere, both with radius $r=6$ mm. The cone's height $h = 14 - 6 = 8$ mm (total height minus hemisphere radius).

Step2: Calculate cone volume

Use cone volume formula $V_{cone}=\frac{1}{3}\pi r^2 h$

$$\begin{align*} V_{cone}&=\frac{1}{3}\pi (6)^2 (8)\\ &=\frac{1}{3}\pi \times 36 \times 8\\ &=96\pi \end{align*}$$

Step3: Calculate hemisphere volume

Use hemisphere volume formula $V_{hemisphere}=\frac{2}{3}\pi r^3$

$$\begin{align*} V_{hemisphere}&=\frac{2}{3}\pi (6)^3\\ &=\frac{2}{3}\pi \times 216\\ &=144\pi \end{align*}$$

Step4: Sum volumes for total

Add cone and hemisphere volumes: $V_{total}=V_{cone}+V_{hemisphere}$
$V_{total}=96\pi + 144\pi = 240\pi$
*Note: Rechecking cone height: total height 14 mm, hemisphere radius 6 mm, so cone height $14-6=8$ mm is correct. Wait, alternative: if total height is cone height (14 mm) plus hemisphere? No, the base of cone and hemisphere are attached, so total height is cone height + hemisphere radius. Wait, re-interpret: if the 14 mm is the cone's height (from tip to base), then:

Corrected Step1: Re-interpret dimensions

Cone height $h=14$ mm, hemisphere radius $r=6$ mm (same as cone base radius)

Corrected Step2: Recalculate cone volume

$$\begin{align*} V_{cone}&=\frac{1}{3}\pi (6)^2 (14)\\ &=\frac{1}{3}\pi \times 36 \times 14\\ &=168\pi \end{align*}$$

Corrected Step3: Hemisphere volume (unchanged)

$V_{hemisphere}=144\pi$

Corrected Step4: Total volume

$V_{total}=168\pi + 144\pi = 312\pi$

Answer:

312π mm³ (Option C: 312π mm³)