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A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains $\frac{1408}{21} \text{ m}^3$ of air, find the height of the cylindrical part. (Use $\pi = \frac{22}{7}$).
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Let $r$ be the radius and $h$ be the height of cylinder. $h = 2r$. Volume $= \frac{2}{3}\pi r^3 + \pi r^2 h = \frac{1408}{21} \implies \frac{2}{3}\pi r^3 + \pi r^2(2r) = \frac{1408}{21} \implies \frac{8}{3} \times \frac{22}{7} \times r^3 = \frac{1408}{21} \implies r^3 = 8 \implies r = 2$ m and $h = 4$ m.