Question

a frustum is formed when a plane parallel to a cones base cuts off the upper portion as shown.
diagram of a cone with height 6.4 and radius 1.5, and a frustum with height 6.8, top radius (implied from the cones radius) and bottom radius 3
what is the volume of the frustum? express the answer in terms of \\(\pi\\).
\\(\bigcirc\\) \\(15.6\pi\\) units\\(^3\\)
\\(\bigcirc\\) \\(20.4\pi\\) units\\(^3\\)
\\(\bigcirc\\) \\(34.8\pi\\) units\\(^3\\)
\\(\bigcirc\\) \\(44.4\pi\\) units\\(^3\\)

Explanation:

Step1: Calculate the volume of the large cone

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). For the large cone, \(r = 3\), \(h=6.4 + 6.8=13.2\).

$$V_{large}=\frac{1}{3}\pi\times3^{2}\times13.2=\frac{1}{3}\pi\times9\times13.2 = 39.6\pi$$

Step2: Calculate the volume of the small cone

For the small cone, \(r = 1.5\), \(h = 6.4\).

$$V_{small}=\frac{1}{3}\pi\times(1.5)^{2}\times6.4=\frac{1}{3}\pi\times2.25\times6.4 = 4.8\pi$$

Step3: Calculate the volume of the frustum

The volume of the frustum \(V = V_{large}-V_{small}\)

$$V=39.6\pi- 4.8\pi=34.8\pi$$

Answer:

\(34.8\pi\) units\(^{3}\)