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

Cevap

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}

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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