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A metallic solid spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 2 cm and 1.5 cm. What is the surface area (in cm2) of the third ball?

A metallic solid spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 2 cm and 1.5 cm. What is the surface area (in cm2) of the third ball? Correct Answer 25 π 

Given :

The radius of the metallic big ball, R = 3 cm

Re-casted in three small balls.

The radius of the first small ball, r1 = 2 cm

The radius of the second small ball, r2 = 1.5 cm

Concept used :

Before and after recasting volume remains the same

Formula used :

The volume of sphere = (4/3) π r3

Calculation :

Let the radius of the third ball be r3

The volume of big ball = sum of the volume of three small balls

⇒ (4/3) π R3 = (4/3) π (r13 + r23 + r33)

⇒ R3 = r13 + r23 + r33

⇒ 23 + 1.53 + r33 = 33

⇒ 8 + 3.375 + r33 = 27

⇒ r33 = 27 - 8 - 3.375

⇒ r33 = 15.625

⇒ r3 = 2.5

Now,

Surface area of sphere of radius 2.5 cm

⇒ Surface area of third sphere = 4 π r32

⇒ Surface area of third sphere = 4 π (2.5)2

⇒ Surface area of third sphere = 4 π  6.25

⇒ Surface area of third sphere = 25 π 

∴ The surface area of third sphere is 25 π cm2.

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