Question

the large sphere has a diameter of 12 feet. what is the volume of the shaded figure? express the answer in terms of π. 252π ft³ 261π ft³ 288π ft³ 324π ft³

Explanation:

Step1: Find radius of large sphere

The diameter of the large sphere is 12 feet, so the radius $R=\frac{12}{2}=6$ feet.

Step2: Recall volume formula for sphere

The volume formula of a sphere is $V = \frac{4}{3}\pi r^{3}$.

Step3: Calculate volume of large sphere

$V_{1}=\frac{4}{3}\pi R^{3}=\frac{4}{3}\pi\times6^{3}=\frac{4}{3}\pi\times216 = 288\pi$ cubic - feet.

Step4: Calculate volume of small sphere

The radius of the small sphere $r = 3$ feet. So $V_{2}=\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi\times3^{3}=\frac{4}{3}\pi\times27 = 36\pi$ cubic - feet.

Step5: Find volume of shaded region

The volume of the shaded region $V=V_{1}-V_{2}=288\pi - 36\pi=252\pi$ cubic - feet.

Answer:

A. $252\pi\ ft^{3}$