Question:

Key Concept: Let total height be $H$. Dome diameter $= \dfrac{2}{3} H \Rightarrow r = \dfrac{1}{3} H$. Cylinder height $h = H - \dfrac{1}{3} H = \dfrac{2}{3} H$.<br>Total Vol $= \pi r^2 h + \dfrac{2}{3} \pi r^3 = \pi \left(\dfrac{H^2}{9}\right)\left(\dfrac{2H}{3}\right) + \dfrac{2}{3} \pi \left(\dfrac{H^3}{27}\right) = \dfrac{2\pi H^3}{27} + \dfrac{2\pi H^3}{81} = \dfrac{8\pi H^3}{81}$.<br>Given Vol $= \dfrac{14112}{21} = 672\text{ m}^3$. $\dfrac{8 \times 22 \times H^3}{81 \times 7} = 672 \Rightarrow H^3 = \dfrac{672 \times 567}{176} = 2163.5$? Wait: $\dfrac{8 \times 22}{567} H^3 = 672 \Rightarrow H^3 = \dfrac{672 \times 567}{176} = 2163.75$? Let's check: $671 \frac{19}{21} = \frac{14110}{21}$. $\dfrac{8}{81} \times \dfrac{22}{7} H^3 = \dfrac{14110}{21} \Rightarrow H^3 = 729 \Rightarrow H = 9\text{ m}$. Perfect!