A frustum of a right circular cone has an upper base radius of 3\text{ cm}, a lower base r… — JAMB UTME

A frustum of a right circular cone has an upper base radius of $3\text{ cm}$ , a lower base radius of $6\text{ cm}$ , and a vertical height of $4\text{ cm}$ . What is the volume of the frustum?

$84\pi\text{ cm}^3$ Answer
B
$252\pi\text{ cm}^3$
C
$60\pi\text{ cm}^3$
D
$81\pi\text{ cm}^3$

Answer

The volume of the frustum is

84\pi\text{ cm}^3

The correct answer is obtained by applying the frustum volume formula

V = \frac{1}{3}\pi h (R^2 + r^2 + R r)

. Substituting

R = 6

r = 3

, and

h = 4

gives

V = \frac{1}{3}\pi(4)(36 + 9 + 18) = 84\pi\text{ cm}^3

Step-by-Step Solution

Identify the given dimensions and formula for the volume of a frustum of a right circular cone.

Upper radius

r = 3\text{ cm}

, lower radius

R = 6\text{ cm}

, height

h = 4\text{ cm}

. Formula:

V = \frac{1}{3}\pi h (R^2 + r^2 + R r)

The volume of a frustum of a cone is derived by subtracting the top small cone from the total original cone.

Evaluate the terms inside the parentheses.

R^2 + r^2 + R r = 6^2 + 3^2 + (6 \times 3) = 36 + 9 + 18 = 63

Computing the effective area scaling factor of the frustum bases.

Multiply by

\frac{1}{3}\pi h

to find the total volume.

V = \frac{1}{3} \times \pi \times 4 \times 63 = 4 \times 21 \times \pi = 84\pi\text{ cm}^3

Completing the frustum volume calculation.

Key Concept

Volume of a Frustum of a Cone

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