Question

the large sphere has a diameter of 20 feet. which expression represents the volume, in cubic units, of the shaded part of the sphere? \\(\frac{4}{3}\pi(10^3) + \frac{4}{3}\pi(4^3)\\) \\(\frac{4}{3}\pi(10^3) - \frac{4}{3}\pi(4^3)\\) \\(\frac{4}{3}\pi(20^3) + \frac{4}{3}\pi(4^3)\\) \\(\frac{4}{3}\pi(20^3) - \frac{4}{3}\pi(4^3)\\)

Explanation:

Step1: Find large sphere radius

The diameter of the large sphere is 20 ft, so radius $r_1 = \frac{20}{2} = 10$ ft.
Volume of large sphere: $V_1 = \frac{4}{3}\pi r_1^3 = \frac{4}{3}\pi(10^3)$

Step2: Find small sphere radius

The small sphere has radius $r_2 = 4$ ft.
Volume of small sphere: $V_2 = \frac{4}{3}\pi r_2^3 = \frac{4}{3}\pi(4^3)$

Step3: Calculate shaded volume

Shaded volume = Large sphere volume - Small sphere volume
$V_{\text{shaded}} = V_1 - V_2$

Answer:

$\boldsymbol{\frac{4}{3}\pi(10^3) - \frac{4}{3}\pi(4^3)}$ (the second option)