Question

a frustum is formed when a plane that is parallel to a cone’s base cuts off the upper portion, as shown below. what is the volume of the frustum? leave the answer in terms of \\(\pi\\). \\(\square\pi\\) units\\(^3\\)

Explanation:

Step1: Recall cone volume formula

The volume of a cone is $V = \frac{1}{3}\pi r^2 h$

Step2: Calculate large cone height

The small cone has height 3, frustum height 6. Total large cone height: $h_{large} = 3 + 6 = 9$

Step3: Find volume of large cone

Substitute $r=4, h=9$:
$V_{large} = \frac{1}{3}\pi (4)^2 (9) = \frac{1}{3}\pi \cdot 16 \cdot 9 = 48\pi$

Step4: Find volume of small cone

Substitute $r=2, h=3$:
$V_{small} = \frac{1}{3}\pi (2)^2 (3) = \frac{1}{3}\pi \cdot 4 \cdot 3 = 4\pi$

Step5: Subtract volumes for frustum

$V_{frustum} = V_{large} - V_{small}$

Answer:

$44\pi$ units$^3$