CONVERSION OF SOLID FROM ONE SHAPE TO ANOTHER EXAMPLES

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%

Example 5 :

A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.

Solution :

Height of cone = 24 cm

radius = 6 cm

Since the cone reshapes into sphere,

volume of cone = volume of sphere

(1/3)πr² h = (4/3)πr³

(6)² 24 = 4r³

r³ = (36 x 24) / 4

r³ = 216

r = 6 cm

So, the radius of the sphere is 6 cm.

Example 6 :

Selvi’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (an underground tank) which is in the shape of a cuboid. The sump has dimensions 1.57 m × 1.44 m × 95cm. The overhead tank has its radius 60 cm and height 95 cm. Find the height of the water left in the sump after the overhead tank has been completely filled with water from the sump which had been full. Compare the capacity of the tank with that of the sump. (Use π = 3.14)

Solution :

The volume of water in the overhead tank equals the volume of the water removed from the sump.

Now, the volume of water in the overhead tank (cylinder)

= πr²h

= 3.14 × 0.6 × 0.6 × 0.95 m³

The volume of water in the sump when full

= l × b × h

= 1.57 × 1.44 × 0.95 m³

The volume of water left in the sump after filling the tank

= [(1.57 × 1.44 × 0.95) – (3.14 × 0.6 × 0.6 × 0.95)] m³

= (1.57 × 0.6 × 0.6 × 0.95 × 2) m³

So, the height of the water left in the sump

= volume of water left in the sump / l b 

= (1.57 × 0.6 × 0.6 × 0.95 × 2) /1.57 × 1.44 × 0.95

= 0.475 m

= 4.75 cm

capacity of tank / capacity of sump

= (3.14 x 0.6 x 0.6 x 0.95) / (1.55 x 1.44 x 0.95)

= 1/2

Therefore, the capacity of the tank is half the capacity of the sump.

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