Example 1 :
Seenu’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (underground tank) which is in the shape of a cuboid. The sump has dimensions 2 m x 1.5 m x 1 m. The overhead tank has its radius of 60 cm and height 105 cm. Find the volume of the water left in the sump after the overhead tank has been completely filled with water from the sump which has been full, initially.
Solution :
Volume o water in sump = Volume of water in cuboidical tank - Volume of water in cylindrical tank
radius of tank = 60 cm = (60/100) m
height of tank = 105 cm = (105/100) m
length = 2 m
width = 1.5 m
height = 1 m
= l ⋅ w ⋅ h - πr2h
= 2 ⋅ 1.5 ⋅ 1 - (22/7)(60/100)2(105/100)
= 3 - 1.188
= 1.812 m3
= 1812000 cm3
Example 2 :
The internal and external diameter of a hollow hemispherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, then find the height of the cylinder.
Solution :
Volume of hollow hemisphere = Volume of cylinder
External radius of hemisphere (R) = 5 cm
Internal radius of hemisphere (r) = 3 cm
Radius of cylinder = 7 cm
(2/3)π(R3 - r3) = πr2h
(2/3)(53 - 33) = 72h
h = (2/3)(125 - 27) / 49
h = 196/147
h = 1.33 cm
Example 3 :
A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 5 cm and its height is 32 cm, then find the thickness of the cylinder.
Solution :
Volume of sphere = Volume of hollow cylinder
(4/3)πr3 = πh(R2 - r2)
(4/3)63 = 32(52 - r2)
25 - r2 = [(4/3) 216]/32
25 - r2 = [(4) 72]/32
25 - r2 = 9
r2 = 25 - 9 = 16
r = 4
Thickness = R - r
= 5 - 4
= 1 cm
Example 4 :
A hemispherical bowl is filled to the brim with juice. The juice is poured into a cylindrical vessel whose radius is 50% more than its height. If the diameter is same for both the bowl and the cylinder then find the percentage of juice that can be transferred from the bowl into the cylindrical vessel.
Solution :
radius of hemispherical bowl = r
Volume of hemispherical bowl = (2/3)πr3
Let "h" be the height of cylindrical vessel
r = h(1 + (50/100))
h = (2r/3)
Volume of cylindrical vessel = πr2h
= πr2(2r/3)
= (2/3)πr3
Hence the answer is 100%.
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