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A vessel in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $ 14 \mathrm{~cm} $ and the total height of the vessel is $ 13 \mathrm{~cm} $. Find the inner surface area of the vessel.
Given:
A vessel in the form of a hollow hemisphere mounted by a hollow cylinder.
The diameter of the hemisphere is \( 14 \mathrm{~cm} \) and the total height of the vessel is \( 13 \mathrm{~cm} \).
To do:
We have to find the inner surface area of the vessel.
Solution:
Diameter of the hollow hemisphere $= 14\ cm$
This implies,
Radius of the hemisphere $=\frac{14}{2}$
$ = 7\ cm$
Total height of the vessel $=13\ cm$
Height of the cylindrical part $=13-7$
$= 6\ cm$
Therefore,
Inner surface area of the vessel $=$ Inner surface area of the cylindrical part $+$ Inner surface area of the hemispherical part
$=2 \pi r h+2 \pi r^{2}$
$=2 \pi r(h+r)$
$=2 \times \frac{22}{7} \times 7(6+7)$
$=2 \times 22 \times 13$
$=572 \mathrm{~cm}^{2}$
The inner surface area of the vessel is $572\ cm^2$.