Question

which expression represents the volume, in cubic units, of the composite figure? \\(\big(\frac{4}{3}\big)\pi(10^3) + \pi(10^2)(28)\\) \\(\big(\frac{4}{3}\big)\pi(20^3) + \pi(20^2)(28)\\) \\(2\big(\frac{4}{3}\big)\pi(10^3) + \pi(10^2)(28)\\) \\(2\big(\frac{4}{3}\big)\pi(20^3) + \pi(20^2)(28)\\)

Explanation:

Step1: Identify figure components

The composite figure is a cylinder with two hemispheres (one at each end; two hemispheres make a full sphere). The diameter is 20, so radius $r = \frac{20}{2}=10$. The cylinder height is 28.

Step2: Volume of the sphere

Volume of a sphere is $V_{sphere}=\frac{4}{3}\pi r^3$. Substitute $r=10$:
$V_{sphere}=\frac{4}{3}\pi(10^3)$

Step3: Volume of the cylinder

Volume of a cylinder is $V_{cylinder}=\pi r^2 h$. Substitute $r=10, h=28$:
$V_{cylinder}=\pi(10^2)(28)$

Step4: Total volume of composite figure

Add the sphere volume and cylinder volume:
$V_{total}=\frac{4}{3}\pi(10^3) + \pi(10^2)(28)$

Answer:

$\boldsymbol{\frac{4}{3}\pi(10^3) + \pi(10^2)(28)}$ (the first option)