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Q) A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area and the volume of the vessel.

Ans.

Let’s start with the diagram of the question: A vessel is in the form of a hollow hemisphere mounted CBSE Question Paper 2024

We are given that the diameter of the hemisphere = 14 cm

Therefore, the radius of the hemisphere = \frac{14}{2} = 7 cm

Next, we can see that the Height of vessel = Height of cylinder + radius of hemisphere

Therefore, Height of cylinder = Height of vessel – radius of hemisphere

= 13 – 7 = 6 cm

(i). Calculating Inner Surface Area:

Total Surface area of vessel = Curved surface area of Cylinder + Surface area of Hemisphere

We know that, Curved surface area of Cylinder = 2 \pi r h and Surface area of Hemisphere = 2 \pi r^2

Therefore, Total Surface area of vessel = 2 \pi r h + 2 \pi r^2 = 2 \pi r (h + r)

Here, r = 7 cm, h = 6 cm

Therefore, Total Surface area of vessel = 2 \pi (7) (6 + 7) = 2 (\frac{22}{7}) (7) (13) = 2 (22) (13) = 572 cm^2

Therefore, Total Surface area of Vessel is 572 cm2 .

Note: Here, thickness of the vessel is not given, hence we neglect the same and calculate total curved surface area of the vessel.

(ii). Calculating Volume of the Vessel:

The volume of the vessel = Volume of the Cylinder + Volume of the Hemisphere

We know that, Volume of Cylinder = \pi r^2 h and Volume of Hemisphere = \frac{2}{3} \pi r^3

Therefore, Volume of the vessel = \pi r^2 h + \frac{2}{3} \pi r^3 = \pi r^2 (h + \frac{2}{3} r)

Here, r = 7 cm, h = 6 cm

Therefore, Volume of the Vessel = (\frac{22}{7}) (7)(7) [6 + (\frac{2}{3})(7)] = 1642.66 cm^3

Therefore, Volume of the vessel is 1642.66 cm3

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