A decorative wooden cone has a slant height of 13\text{ cm} and a vertical height of 12\te… — JAMB UTME

A decorative wooden cone has a slant height of $13\text{ cm}$ and a vertical height of $12\text{ cm}$ . What is the volume of the cone in terms of $\pi$ ?

$100\pi\text{ cm}^3$ Answer
B
$300\pi\text{ cm}^3$
C
$400\pi\text{ cm}^3$
D
$65\pi\text{ cm}^3$

Answer

The volume of the cone is

100\pi\text{ cm}^3

The radius is determined via the Pythagorean relation

r = \sqrt{13^2 - 12^2} = 5\text{ cm}

. Substituting

r = 5\text{ cm}

and

h = 12\text{ cm}

into

V = \frac{1}{3}\pi r^2 h

yields

V = \frac{1}{3}\pi(25)(12) = 100\pi\text{ cm}^3

Step-by-Step Solution

Find the base radius of the cone using the Pythagorean theorem.

r = \sqrt{l^2 - h^2} = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5\text{ cm}

The radius, vertical height, and slant height of a right circular cone form a right-angled triangle where the slant height is the hypotenuse.

Calculate the volume of the cone using the formula

V = \frac{1}{3}\pi r^2 h

V = \frac{1}{3} \times \pi \times 5^2 \times 12 = \frac{1}{3} \times \pi \times 25 \times 12 = 100\pi\text{ cm}^3

The formula for the volume of any right circular cone requires multiplying one-third of the base area by the vertical height.

Key Concept

Volume of a Right Circular Cone

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