Question

An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is $\frac{1}{4}$ of the radius of the original ball, how many such balls are made? Compare the surface area, of all the smaller balls combined together with that of the original ball.

Answer 1

Answer:

Number of balls made  = 64  and the total surface area of small balls is equal to 4 times surface area of original ball.

Step-by-step explanation:

Let radius of smaller ball be r.  

Given : Radius of each of smaller ball = 1/4 Radius of original ball.

r = ¼ × Radius of original ball.

Radius of original ball, r1 =  4r

Volume of original ball ,V1 = 4/3 πr1³

Volume of original ball ,V1 = 4/3 π(4r)³

Volume of Smaller ball , V2= 4/3 π(r)³

Number of balls = Volume of original ball / Volume of Smaller ball

Number of balls = 4/3 π(4r)³/ 4/3 π(r)³

= 64r³/r³ = 64 / 1

Number of balls = 64

Surface area of sphere = 4πr2

Surface area of original ball (S1) = 4π(4r)2      

Surface area of each Smaller ball ,(S2) = 4πr2

Total surface area of 64 small balls , S2 = 64 × 4πr2

Total surface area of 64 small balls / Surface area of original ball = 64 × 4πr² / 4π(4r)2

S2/ S1 = (64 × 4πr²) / (4π × 16r²)

S2/ S1 = 64/16 = 4  

S2/ S1 = 4

S2 : S1 = 4 : 1

S2 = 4S

Hence,the total surface area of small balls is equal to 4 times surface area of original ball.

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Answer 2

ANSWER:--

Given : Radius of each of smaller ball = 1/4

Radius of original ball.

r = ¼ × Radius of original ball. Radius of original ball, r1 =  4r

Volume of original ball ,V1 = 4/3 πr1³ Volume of
original ball ,V1 = 4/3 π(4r)³ Volume of

Smaller ball , V2= 4/3 π(r)³ Number of balls =

Volume of original ball /
Volume of Smaller ball

Number of balls = 4/3 π(4r)³/ 4/3 π(r)³ = 64r³/r³
= 64 / 1 Number of balls
= 64 Surface area of

sphere = 4πr2 Surface area of original ball (S1)
= 4π(4r)2      Surface area of each Smaller ball ,(S2)

= 4πr2 Total surface area of 64 small balls , S2

= 64 × 4πr2 Total surface area of 64 small balls / Surface area of original ball

= 64 × 4πr² / 4π(4r)2 S2/ S1

= (64 × 4πr²) / (4π × 16r²) S2/ S1

= 64/16
= 4  S2/ S1
= 4 S2 : S1 = 4 : 1 S2 = 4S Hence,the total

surface area of small balls is equal to 4 times surface area of original ball.

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