Question

If 125 identical metallic balls are melted to form 8 bigger balls, each of same size, then what is the ratio of the surface areas of a bigger ball to that of a smaller ball​?

a) 4:3
b) 5:2
c) 16:9
d) 25:4​

Answer 1

Answer:

Let , radius of spherical ball be R,

And radius of 8 identical balls be r,

According to question,

Volume of larger sphere =8×volume of 8 identical balls

34πR 3 =8× 34πr 3⟹R 3 =8×r 3

Hence, R=2r

Answer 2

Let :

• Radius of small ball = x
• Radius of bigger ball = y

Given :

125 small ball are melted to form 8 big ball

To find :

Ratio of surface area of bigger ball to smaller ball

Formula used :

• Volume of sphere = (4/3)πr³
• Surface area of sphere = 4πr²

Solution :

125 small ball are melted to form 8 big ball

Therefore volume of 125 smaller ball = Volume of 8 bigger ball

$\sf \implies \: 125 \times \bigg( \dfrac{4}{3} \times \pi \times {r}^{3} \: \bigg) = \: 8 \times \bigg(\dfrac{4}{3} \times \pi \times {R}^{3} \bigg) \\ \\ \sf \implies \: \bigg( \dfrac{ {r}^{3} }{ {R}^{3}} \bigg) \: = \: \dfrac{8}{125} \\ \\ \sf \implies \: \bigg( \dfrac{ {r} }{ {R}} \bigg)^{3} \: = \: \dfrac{ {2}^{ 3 } }{ {5}^{3} } \\ \\ \sf \implies \: \bigg( \dfrac{ {r} }{ {R}} \bigg)^{3} \: = \: \bigg( \dfrac{ {2} }{ {5}} \bigg)^{3} \\ \\ \sf \: take \: cube \: root \: on \: both \: side \\ \sf \implies \: \dfrac{ {r} }{ {R}} \: = \: \dfrac{ {2} }{ {5}}$

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Now , surface area of sphere = 4πr²

➝ Surface area of bigger ball : Surface area of smaller ball = 4π (R²) : 4π r²

$\sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \: \dfrac{4π R^{2} }{4\pi \: {r}^{2} } \\ \\ \sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \: \: \dfrac{R^{2}}{ {r}^{2} } \\ \\ \sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \: \bigg(\dfrac{R}{ {r}} \bigg)^{2} \\ \\ \sf \: put \: value \: of \: \dfrac{R}{r} \\\sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \: \bigg(\dfrac{5}{ {2}} \bigg)^{2} \\ \\ \sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \: \dfrac{ {5}^{2} }{ {2}^{2} } \\ \\ \sf \implies \: \dfrac{Surface \: area \: of \: bigger \: ball }{Surface \: area \: of \: smaller \: ball } \: = \dfrac{25}{4}$

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ANSWER :

Option d) 25:4