Question

A solid consisting of a right circular cone standing on a hemisphere, is placed upright in a right circular cylinder, full of water and touches the bottom. Find the volume of water left in the cylinder having given that the radius of the cylinder is 3 cm and its height is 6 cm. The radius of hemisphere is 2 cm and the height of the cone is 4 cm. ( Also draw the diagram)

Answer 1

Let, height of cylinder be H and radius R; height of cone be h and radius of cone
and hemisphere be r
Volume of water left in the cylinder = [Volume of cylinder – (Volume of cone
+ Volume of hemisphere) ]
$V=\pi R^{2}H–\left[\right(\frac{1}{3})\pi r^{2}h+\frac{2}{3}\pi r^3]$
Or, $V=(\pi \times 3^2\times 6)–(\frac{1}{3}\times \pi \times 2^2\times 4+\frac{2}{3}\times \pi \times 2^3)$
Or, $V=54\pi –(\frac{16\pi }{3}+\frac{16\pi }{3})$
Or, $V=[54\pi –\frac{32\pi }{3}]$
Or, $V=\frac{(162\pi –32\pi )}{3}$
Or, $V=\left(\frac{130}{3}\right)(\frac{22}{7}$)
Or, $V=136c{m}^3$