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 of the vessel.
Step-by-Step Solution
Key Concept: The inner surface area of the vessel is the sum of the curved surface area of the hollow hemisphere and the curved surface area of the hollow cylinder. Use the formulas: \(\text{Curved surface area of a hemisphere}=2\pi r^{2}\) and \(\text{Curved surface area of a cylinder}=2\pi r h\).
1. Find the radius of the hemisphere\
The diameter is 14 cm, therefore\
\[ r = \frac{\text{diameter}}{2}=\frac{14}{2}=7\ \text{cm}. \]
2. Determine the height of the cylindrical part\
The total height of the vessel = height of hemisphere + height of cylinder.\
Height of hemisphere = radius = 7 cm.\
Hence,\
\[ h_{\text{cyl}} = 13\ \text{cm} - 7\ \text{cm}=6\ \text{cm}. \]
3. Curved surface area of the hollow hemisphere\
\[ \text{CSA}_{\text{hemisphere}} = 2\pi r^{2}=2\pi (7)^{2}=2\pi \times 49 = 98\pi\ \text{cm}^{2}. \]
4. Curved surface area of the hollow cylinder\
\[ \text{CSA}_{\text{cylinder}} = 2\pi r h = 2\pi (7)(6)=84\pi\ \text{cm}^{2}. \]
5. Total inner surface area\
\[ \text{Total inner surface area}= \text{CSA}_{\text{hemisphere}}+\text{CSA}_{\text{cylinder}}\]
\[ = 98\pi + 84\pi = 182\pi\ \text{cm}^{2}. \]
6. Numerical value (optional)\
Using \(\pi \approx \frac{22}{7}\),\
\[ 182\pi \approx 182 \times \frac{22}{7}=572\ \text{cm}^{2}. \]
Or with \(\pi \approx 3.14\),\
\[ 182\pi \approx 182 \times 3.14 = 571.48\ \text{cm}^{2}. \]
Thus, the inner surface area of the vessel is \(182\pi\ \text{cm}^{2}\) (approximately \(572\ \text{cm}^{2}\)).
Correct Answer: \(182\pi\ \text{cm}^{2}\) (≈ 572 cm²)