A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylinder is $\frac{14}{3} \mathrm{~m}$ and the diameter of hemisphere is $3.5 \mathrm{~m}$. Calculate the volume and the internal surface area of the solid.
Given that:
Radius of the same base $r=\frac{3.5}{2}=1.75 \mathrm{~m}$
Height of the cylinder $h=\frac{14}{3} \mathrm{~m}$
The volume of the vessel is given by
$V=\pi r^{2} h+\frac{2}{3} \pi r^{3}$
$=3.14 \times 1.75^{2} \times \frac{14}{3}+\frac{2}{3} \times 3.14 \times 1.75^{3}$
$=56 \mathrm{~m}^{3}$
The internal surface area of the solid is
$S=2 \pi r^{2}+2 \pi r h$
$=2 \times 3.14 \times 1.75^{2}+2 \times 3.14 \times 1.75 \times \frac{14}{3}$
$=70.51 \mathrm{~m}^{2}$
Hence, the volume of the vessel and internal surface area of the solid is $V=56 \mathrm{~m}^{3}, S=70.51 \mathrm{~m}^{2}$