Question 1 :
An aluminium sphere of radius 12 cm is melted to make a cylinder of radius 8 cm. Find the height of the cylinder.
Solution :
Volume of sphere = Volume of cylinder
(4/3) πr3 = πr2h
Radius of sphere = 12 cm
Radius of cylinder = 8 cm
Height of cylinder = h
(4/3) 123 = 82h
h = (4/3) (12⋅12⋅12) / (8⋅8)
h = 36 cm
Question 2 :
Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level of water in the tanks will rise by 21 cm.
Solution :
Quantity of water flowing out = Quantity of water in rectangular tank
Speed of water = 15 km per hour
= 15000 m
Diameter of pipe = 14 cm = (14/100) m
radius of pipe = 7/100
length of rectangular tank = 50 m,
width of tank = 44 m
height of tank = 21 cm = (21/100) m
Area of cross section ⋅ Time ⋅ Speed = l ⋅ w ⋅ h
πr2 ⋅ Time ⋅ 15000 = (50) ⋅ (44) ⋅ (21/100)
(22/7) ⋅ (7/100)2 ⋅ Time ⋅ 15000 = (50) ⋅ (44) ⋅ 21
Time = (50⋅44⋅21⋅100⋅100⋅7)/(15000⋅100⋅7⋅ 7⋅22)
time = 2 hours
Question 3 :
A conical flask is full of water. The flask has base radius r units and height h units, the water poured into a cylindrical flask of base radius xr units. Find the height of water in the cylindrical flask
Solution :
Volume of water poured from the conical tank = Volume of water in cylindrical flask
(1/3)πr2h = πr2h
radius of cylinder = xr
(1/3)πr2h = π(xr)2h
(r2h/3) = (x2r2)h
h = r2h/3x2r2
height of cylinder = h/3x2
Question 4 :
A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter.
Solution :
radius of cone = 7 cm
height of cone = 8 cm
Radius of sphere = 5 cm
internal radius = ?
Volume of cone = Volume of sphere
(1/3)πr2h = (4/3)π(R3 - r3)
72(8) = 4 (53 - r3)
(125 - r3) = 49(2)
125 - r3 = 98
r3 = 125 - 98
r3 = 27
r = 3
Hence the required internal radius is 3 cm.
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