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 $\frac{1400}{3}$ m$^3$ of

Question

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 $\frac{1400}{3}$ m$^3$ of air, find the height of the cylindrical part. (Use $\pi = \frac{22}{7}$)

Accepted Answer

Step 1: Let $r$ be the radius of hemisphere and $h$ be the height of cylinder. Given $h = 2r$.
Step 2: Volume of air $= \frac{2}{3}\pi r^3 + \pi r^2 h = \frac{2}{3}\pi r^3 + \pi r^2(2r) = \frac{2}{3}\pi r^3 + 2\pi r^3 = \frac{8}{3}\pi r^3$
Step 3: $\frac{1400}{3} = \frac{8}{3} 	imes \frac{22}{7} 	imes r^3 \Rightarrow r^3 = \frac{1400 	imes 7}{8 	imes 22} = \frac{9800}{176} = \frac{1225}{22}$
Step 4: $r^3 = 8 \Rightarrow r = 2$ m and $h = 2r = 4$ m

Question

Solution