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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.
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Radius = $7$ cm Height of cylindrical portion = $13 - 7 = 6$ cm Inner surface area of the vessel = $2\pi r^2 + 2\pi rh = 2 \times \frac{22}{7} \times 7 \times 7 + 2 \times \frac{22}{7} \times 7 \times 6 = 572 \text{ cm}^2$ Volume of the vessel $= \frac{2}{3} \pi r^3 + \pi r^2h = \frac{2}{3} \times \frac{22}{7} \times 7 \times 7 \times 7 + \frac{22}{7} \times 7 \times 7 \times 6 = \frac{4928}{3}$ or $1642.67 \text{ cm}^3$ approx. Therefore, inner surface area and volume of the vessel is $572 \text{ cm}^2$ and $1642.67 \text{ cm}^3$ respectively.