A solid trophy consists of a right circular cone mounted on top of a right circular cylind… — JAMB UTME

A solid trophy consists of a right circular cone mounted on top of a right circular cylinder with the same base radius of $6\text{ cm}$ . The height of the cylinder is $10\text{ cm}$ and the slant height of the cone is $10\text{ cm}$ . What is the total volume of the trophy?

A
$216\pi\text{ cm}^3$
$456\pi\text{ cm}^3$ Answer
C
$480\pi\text{ cm}^3$
D
$648\pi\text{ cm}^3$

Answer

The total volume of the trophy is

456\pi\text{ cm}^3

First, use the Pythagorean theorem on the cone to find its vertical height

h_{\text{cone}} = \sqrt{10^2 - 6^2} = 8\text{ cm}

. The volume of the cone is

\frac{1}{3}\pi (6)^2 (8) = 96\pi\text{ cm}^3

. The volume of the cylinder is

\pi (6)^2 (10) = 360\pi\text{ cm}^3

. Adding both gives a total volume of

456\pi\text{ cm}^3

Step-by-Step Solution

Calculate the vertical height of the cone

h_{\text{cone}} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8\text{ cm}

The volume formula for a cone requires the vertical height (

h

), which forms a right-angled triangle with the radius (

r=6\text{ cm}

) and slant height (

l=10\text{ cm}

Calculate the volume of the conical part

V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}} = \frac{1}{3} \times \pi \times 6^2 \times 8 = 96\pi\text{ cm}^3

The formula for the volume of a right circular cone is

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

Calculate the volume of the cylindrical part

V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}} = \pi \times 6^2 \times 10 = 360\pi\text{ cm}^3

The formula for the volume of a right circular cylinder is

V = \pi r^2 h

Sum the volumes of both 3D shapes to get the total volume

V_{\text{total}} = 96\pi + 360\pi = 456\pi\text{ cm}^3

The total volume of a composite solid is the sum of the volumes of its constituent parts.

Key Concept

Volume of Composite 3D Solids

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