A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are $ 5 \mathrm{~cm} $ and $ 13 \mathrm{~cm} $ respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Find the surface area of the toy if the total height of the toy is $ 30 \mathrm{~cm} $.

Given:

A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other.

The radius and height of the cylindrical part are \( 5 \mathrm{~cm} \) and \( 13 \mathrm{~cm} \) respectively.

The radii of the hemispherical and conical parts are the same as that of the cylindrical part.

The total height of the toy is \( 30 \mathrm{~cm} \).

To do:

We have to find the surface area of the toy.

Solution:

Radius of the base of the cylindrical part $r= 5\ cm$

Height of the cylindrical part $h_1 = 13\ cm$

Height of the conical part $h_2 = 30 - (13 + 5)$

$= 30-18$

$= 12\ cm$

Slant height of the conical part $l=\sqrt{r^{2}+h_{2}^{2}}$

$=\sqrt{(5)^{2}+(12)^{2}}$

$=\sqrt{25+144}$

$=\sqrt{169}$

$=13 \mathrm{~cm}$
Radius of the hemispherical part $r=5 \mathrm{~cm}$

Total surface area of the toy $=$ Curved surface area of the conical part $+$ Curved surface area of the cylindrical part $+$ Curved surface area of the hemisphere

$=\pi r l+2 \pi r h_{1}+2 \pi r^{2}$

$=\pi r(l+2 h_{1}+2 r)$

$=\frac{22}{7} \times 5[13+2 \times 13+2 \times 5]$

$=\frac{110}{7}[13+26+10]$

$=\frac{110}{7} \times 49$

$=770 \mathrm{~cm}^{2}$

The surface area of the toy is $770\ cm^2$.

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Updated on: 10-Oct-2022

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