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.

Solution

Let r cm be the radius and h cm be the height of a cone, then
r = 60 cm, h = 120 cm
Volume of cone =

1 third space πr squared straight h

open parentheses 1 third straight pi space straight x space 60 space straight x space 60 space straight x space 120 close parentheses space c m cubed equals space 144000 space straight pi space cm cubed space

Let r₁ cm be the radius of a hemisphere then
r₁ = 60 cm
Now,
Volume of hemisphere =

2 over 3 πr cubed

equals space open parentheses 2 over 3 straight pi space straight x space 60 space straight x space 60 space straight x space 60 close parentheses space cm cubed equals space 144000 space straight pi space cm cubed

Let R cm be the radius and H cm be the height of a cyclinder, then
R = 60 cm, H = 180 cm
Now,
Volume of Cylinder = πR²H
= (π x 60 x 60 x 180) cm³= (π x 64800) cm³= 648000 π cm³.
Volume of Solid
= Volume of cone + volume of hemisphere
= (144000π + 144000π) cm³= 288000π cm³ Hence,
Volume of water left in the cylinder
= Volume of cylinder – Volume of solid
= 1648000 π – 288000 π) cm³= 360000 π cm³
$equals space open parentheses 360000 space straight x space 22 over 7 close parentheses space cm cubed equals space 7920000 over 7000000 space equals space 1.131 space straight m cubed$

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