VOLUME OF COMBINATION OF SOLIDS PROBLEMS

Problem 1 :

A hollow cylindrical pipe is of length 40 cm. Its internal and external radii are 4 cm and 12 cm respectively. It is melted and cast into a solid cylinder of length 20 cm. Find the radius of the new solid.

Solution :

length of cylindrical pipe (h) = 40 cm

internal radius of the pipe (r) = 4 cm

external radius of the pipe (R) = 12 cm

Height of cylinder = 20 cm

Volume of hollow cylindrical pipe = Volume of cylinder

Π h (R² - r²) =  Π r² h

h (R² - r²)  =   r² h

(40) (12² - 4²)  =  r² (20)

(40) (144 - 16) =  r² (20)

(40) (128)/20  =  r²

r²  =  (40) (128)/20

r²  =  2 (128)

r  =  √256

r  =  16 cm

Therefore radius of cylinder  =  16 cm

Problem 2 :

An iron right circular cone of diameter 8 cm and height 12 cm is melted and recast into spherical lead shots each of radius 4 mm. How many lead shots can be made?

Solution :

diameter of right circular cone = 8 cm

radius of right circular cone (r) = 4 cm

Height of right circular cone (h) = 12 cm

Radius of spherical lead shot (r) = 4 mm

10 mm  =  1 cm

4 mm  =  (4/10) cm 

Volume of right circular cone

= n x (Number of spherical lead shots)

n  =  Volume of cone/Number of spherical lead shots

n  =   (1/3)Π r² h/(4/3)Π r³

n  =  (1/3) (4)² (12) x (3/4) (10/4)³

n =  3 x 5 x 5 x 10

n = 750 lead shots

Therefore 750 lead shots can be made.

Problem 3 :

A container with a rectangular base of length 4.4 m and breadth 2 m is used to collect rain water. The height of the water level in the container is 4 cm and water is transferred into a cylindrical vessel with radius 40 cm. What will be the height of the water level in the cylinder?

Solution :

Volume of water in cuboidal container  =  Volume of water in cylindrical container

lwh  =  Π r²h

l  =  4.4 m, w  =  2 m, h  =  4 cm ==>  4/100 ==>  1/25 m

r  =  40 cm ==>  40/100

4.4(2) (1/25)  =  Π (2/5)²h

h  =  4.4 (2) (1/25) (1/Π) (25/4)

h  =  2.2/Π

h  =  0.7 m

Problem 4 :

A cylindrical bucket of height 32 cm and radius 18 cm is filled with sand. The bucket is emptied on the ground and a conical heap and sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.

Solution :

Volume of sand in the cylindrical bucket  =  Volume of sand of conical heap

Let r₁, h₁ be the radius and height of cylindrical tank

Let r₂, h₂ be the radius and height of conical tank

Π r₁²h₁  =  (1/3)Π r₂²h₂

r₁  =  18 cm,  h₁  =  32 cm and h₂  =  24 cm

18²(32)  =  (1/3) r₂²(24)

18²(32)  =  8 r₂²

r₂² =  (18⋅18⋅32)/8

r₂  =  18(2)

r₂  =  36 cm

So, the radius of the conical heap is 36 cm.

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