Question

A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14 cm and the total height of the vessel is 13 cm. Find the total surface area of the vessel. (Use $\pi ={\displaystyle \frac{22}{7}}$)

Answer 1

Hint: Here, we will proceed by using the formula i.e., Curved surface area of cylinder = $2\pi rh$ where the radius is found using the formula i.e., Radius = ${\displaystyle \frac{\text{Diameter}}{2}}$. Then, we will apply the formula i.e., Curved surface area of the hemisphere = $2\pi r^2$.

Complete step-by-step answer:
Given, Diameter of the hemispherical bowl, d = 14 cm
Total height of the vessel, H = 13 cm
It is also given that the diameter of the hollow cylinder and that of the hemispherical bowl are same
i.e., Diameter of the hollow cylinder = Diameter of the hemispherical bowl = 14 cm
Let the radius and the height of the hollow cylinder in the vessel be r cm and h cm
Using the formula Radius = ${\displaystyle \frac{\text{Diameter}}{2}}$, we can write
Radius of the hollow cylinder = Radius of the hemispherical bowl = r = ${\displaystyle \frac{\text{14}}{2}}$ = 7 cm
Clearly, the height of the hemispherical bowl portion of the height is equal to the radius of the hemispherical bowl.
i.e., Height of the hemispherical bowl = r = 7 cm
From the figure, we can write
Height of the hollow cylinder = Total height of the vessel – Height of the hemispherical bowl
$\Rightarrow$h = 13 – 7 = 6 cm
As we know that the curved surface of any cylinder having radius r and height h is given by
Curved surface area of cylinder = $2\pi rh$
Using the above formula and taking $\pi ={\displaystyle \frac{22}{7}}$, we have
Curved surface area of the hollow cylinder = $2\left({\displaystyle \frac{22}{7}}\right)\left(7\right)\left(6\right)=264\text{ c}{\text{m}}^2$
Also we know that the curved surface area of any hemisphere with radius as r is given by
Curved surface area of the hemisphere = $2\pi r^2$
Using the above formula and taking $\pi ={\displaystyle \frac{22}{7}}$, we can write
Curved surface area of the given hemispherical bowl = $2\left({\displaystyle \frac{22}{7}}\right){\left(7\right)}^2=308\text{ c}{\text{m}}^2$
Since, the vessel is completely hollow so we can write,
Total surface area of the vessel = 2(Curved surface area of the hollow cylinder + Curved surface area of the given hemispherical bowl)
$\Rightarrow$Total surface area of the vessel = 2(264+308) = 2(572) = 1144 $\text{c}{\text{m}}^2$

Note- In this particular problem, the complete vessel is hollow because of which the total surface area of the vessel will constitute the inner as well as the outer curved surface area of the vessel. Also, the wall thickness of the vessel is assumed to be very thin so that we can neglect it which will give the inner dimensions equal to the outer dimensions.