Mensuration-61

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A spherical ball of lead 3 cm in diameter is melted and recast into 3 spherical balls. The diameters
of the two of these are 1 cm and 1.5 cm respectively. What is the diameter of the third spherical ball?

Solution: Radius of the sphere = 3/2 cm

Volume of the sphere = 4/3 Π r3 = 4/3 x 22/7 x (3/2)3 = 99/7 cm3

Diameter of the 1st small sphere = 1 cm; Therefore, radius = 1/2 cm

Volume of the 1st small sphere = 4/3 Π r3 = 4/3 x 22/7 x (1/2)3 = 11/21 cm3

Diameter of the 2nd small sphere = 1.5 cm; Therefore, radius = 1.5/2 = 3/4 cm

Volume of the 2nd small sphere = 4/3 Π r3 = 4/3 x 22/7 x (3/4)3 = 99/56 cm3

Total volume of the two small spheres = 11/21 + 99/56 = (88 + 297) / 168 = 385 / 168 cm3

Volume of the 3rd small sphere = (99 / 7 – 385 / 168) cm3 = (2376 – 385) / 168 cm3 1991 / 168 cm3

Let the radius of the 3rd small sphere be r cm. Then

4/3 Π r3 = 1991 / 168

Implies, r3 = 1991 / 168 x 3/4 x 7 / 22 = 181 / 64

Implies, r = (181)1/3 / 4 cm

Hence, diameter of the 3rd small spherical ball = 2r = 2 x 1/4 x (181)1/3 cm = 1/2(181)1/3 cm

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