A solid hemisphere of radius 6\text{ cm} has the same total surface area as a solid right … — JAMB UTME

A solid hemisphere of radius $6\text{ cm}$ has the same total surface area as a solid right circular cone with a base radius of $6\text{ cm}$ . What is the slant height of the cone?

A
$6\text{ cm}$
$12\text{ cm}$ Answer
C
$18\text{ cm}$
D
$24\text{ cm}$

Answer

The slant height of the cone is

12\text{ cm}

The correct answer is derived by setting the total surface area of the solid hemisphere (

3\pi R^2 = 108\pi\text{ cm}^2

) equal to the total surface area of the solid cone (

\pi r^2 + \pi r l = 36\pi + 6\pi l

). Solving

36\pi + 6\pi l = 108\pi

gives

l = 12\text{ cm}

Step-by-Step Solution

Calculate the total surface area of the solid hemisphere.

\text{TSA}_{\text{hemisphere}} = 3\pi R^2 = 3\pi(6)^2 = 108\pi\text{ cm}^2

A solid hemisphere consists of a curved surface (

2\pi R^2

) plus its flat circular base (

\pi R^2

Express the total surface area of the solid cone in terms of slant height

l

\text{TSA}_{\text{cone}} = \pi r^2 + \pi r l = \pi(6)^2 + \pi(6)l = 36\pi + 6\pi l

A solid cone has a base area of

\pi r^2

and a curved surface area of

\pi r l

Equate the two surface area expressions and solve for

l

36\pi + 6\pi l = 108\pi \implies 6\pi l = 72\pi \implies l = 12\text{ cm}

The question states that the total surface area of both solids is equal.

Key Concept

Total Surface Area of Composite 3D Solids (Hemisphere and Cone)

Question

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Step-by-Step Solution

Key Concept

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