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A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylinder is $ \frac{14}{2} \mathrm{~m} $ and the diameter of hemisphere is $ 3.5 \mathrm{~m} $. Calculate the volume and the internal surface area of the solid
Given:
A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base.
The depth of the cylinder is \( \frac{14}{2} \mathrm{~m} \) and the diameter of hemisphere is \( 3.5 \mathrm{~m} \).
To do:
We have to find the volume and the internal surface area of the solid.
Solution:
Diameter of the hollow cylinder $= 3.5\ m$
This implies,
Radius of the cylinder $r = \frac{3.5}{2}$
$ =\frac{7}{4}\ m$
Height of the cylindrical part $h = \frac{14}{3}\ m$
Volume of the solid $=$ Volume of the cylindrical part $+$ Volume of the hemispherical part
$=\pi r^{2} h+\frac{2}{3} \pi r^{3}$
$=\pi r^{2}(h+\frac{2}{3} r)$
$=\frac{22}{7} \times(\frac{7}{4})^{2}(\frac{14}{3}+\frac{2}{3} \times \frac{7}{4})$
$=\frac{22 \times 49}{7 \times 16}(\frac{14}{3}+\frac{7}{6})$
$=\frac{77}{8}(\frac{28+7}{6})$
$=\frac{77}{8} \times \frac{35}{6}$
$=\frac{2695}{48}$
$=56.1458$
$=56.15 \mathrm{~m}^{3}$
Internal surface area of the solid $=2 \pi r h+2 \pi r^{2}$
$=2 \pi r(h+r)$
$=2 \times \frac{22}{7} \times \frac{7}{4}(\frac{14}{3}+\frac{7}{4})$
$=11(\frac{56+21}{12})$
$=11 \times \frac{77}{12}$
$=\frac{847}{12}$
$=70 \frac{7}{12} \mathrm{~m}^{2}$
The volume and the internal surface area of the solid are $56.15 \mathrm{~m}^{3}$ and $70 \frac{7}{12} \mathrm{~m}^{2}$.