Question

A solid consisting of a right circular cone of height $120$ cm and radius $60$ cm standing on a hemisphere of radius $60$ cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60$ cm and its height is $180$ cm.

Answer 1

Given:
Radius ($r$) of hemispherical part = Radius ($r$) of conical part $=60$cm
Height ($h_1$) of conical part of solid $=120$ cm
Height ($h_1$) of cylinder $=180$ cm
Radius ($r$) of cylinder $=60$ cm

Volume of water left = Volume of cylinder - Volume of solid

= Volume of cylinder - (Volume of cone + Volume of hemisphere)

$=\pi r^2{h}_1-[\frac{1}{3}\pi r^2{h}_2+\frac{2}{3}\pi r^3]$

$=\pi \times {60}^2\times \left(180\right)-[\frac{1}{3}\pi \times {60}^2\times 120+\frac{2}{3}\pi \times {60}^3]$

$=\pi \left(60{)}^2\right[180-(40+40)]$

$=\pi \left(3600\right)\left[100\right]$

$=3,60,000\times \frac{22}{7}=1131428.57$ cm${}^3$