Question

which expression represents the volume, in cubic units, of the composite figure?
$\bigcirc\\ \pi(5^2)(13) - \frac{1}{3}\pi (5^2)(12)$
$\bigcirc\\ \pi(5^2)(13) - \frac{1}{3}\pi (5^2)(25)$
$\bigcirc\\ \pi(5^2)(13) + \frac{1}{3}\pi (5^2)(12)$
$\bigcirc\\ \pi(5^2)(13) + \frac{1}{3}\pi (5^2)(25)$

Explanation:

Step1: Find cone height

The total height of the figure is 25, and the cylinder height is 13. Subtract to get the cone height:
$25 - 13 = 12$

Step2: Cylinder volume formula

The cylinder has radius $r=5$, height $h=13$. Volume of a cylinder is $V_{cyl} = \pi r^2 h$:
$V_{cyl} = \pi(5^2)(13)$

Step3: Cone volume formula

The cone has radius $r=5$, height $h=12$. Volume of a cone is $V_{cone} = \frac{1}{3}\pi r^2 h$:
$V_{cone} = \frac{1}{3}\pi(5^2)(12)$

Step4: Total composite volume

Add the cylinder and cone volumes together:
$V_{total} = \pi(5^2)(13) + \frac{1}{3}\pi(5^2)(12)$

Answer:

$\boldsymbol{\pi(5^2)(13) + \frac{1}{3}\pi (5^2)(12)}$ (the third option)