Question

A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of the same diameter. The diameter of the hemispherical bowl is 14cm and the total height of the vessel is 13cm. Find the total surface area of the vessel.

Answer 1

Hint: For solving this question, the diameter of the hemispherical bowl is given and using the diameter we can easily calculate the radius. By subtracting the radius of the hemisphere from the height of vessels, we get the height of the cylinder. By using the formula for surface areas, we calculate the total surface area of the vessel.

Complete step-by-step solution -

The diameter of the hemispherical bowl = 14 cm
Radius of the hemispherical bowl $=r=\dfrac{14}{2}=7cm$
Total height of the vessel = 13 cm
$\therefore $ Height of the cylinder, h = height of vessel – radius of cylinder.
$h=13cm-7cm=6cm$
Since it is a hollow vessel so the total surface area is the sum of inner and outer surfaces of the vessel.
Total surface area =2 (Curved surface area of the cylinder + curved surface of hemisphere)
As, curved surface area of cylinder = $2\pi rh$.
Curved surface area of hemisphere = $2\pi {{r}^{2}}$.
Total surface area $=2\left( 2\pi rh+2\pi {{r}^{2}} \right)=4\pi r\left( h+r \right)$
Total surface area $=4\times \dfrac{22}{7}\times 7\times \left( 6+7 \right)c{{m}^{2}}=1144c{{m}^{2}}$
Hence, the total surface area of the vessel is $1144c{{m}^{2}}$.

Note: Students must remember that we are given the total height of the vessel in the problem statement. So, to obtain the net height of the hollow cylindrical portion, we must subtract the radius of the hemispherical bowl from total height. The key point to solve this problem is knowledge of the curved surface area of the cylinder and curved surface area of the hemisphere.