A solid consisting of a right circular cone of height 120 cm

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 bottoms. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

Solution:

We have to find the remaining volume of water left in the cylinder when the solid is inserted into it. The solid is a hemisphere surmounted by a cone.

Radius of cone, cylinder and hemisphere $(r)=60 \mathrm{~cm}$

Height of cone $(l)=120 \mathrm{~cm}$

Height of the cylinder $(h)=180 \mathrm{~cm}$

So the remaining volume of water left in the cylinder when the solid is inserted into it,

$=\pi r^{2} h-\left(\frac{1}{3} \pi r^{2} l+\frac{2}{3} \pi r^{3}\right)$

$=\pi r^{2}\left(h-\frac{1}{3} l-\frac{2}{3} r\right)$

Put the values to get,

$=\left(\frac{22}{7}\right)(3600)(180-40-40) \mathrm{m}^{3}$

$=1.131 \mathrm{~m}^{3}$

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