CONVERSION OF SOLIDS FROM ONE SHAPE TO ANOTHER PRACTICE QUESTIONS

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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