Question

$27$ metal balls each of radius r are melted together to form one big sphere of radius R. Then the ratio of surface area of the big sphere to that of a ball is __________.

• A. $9:1$
• B. $3:2$
• C. $\sqrt{27}:\sqrt{5}$
• D. $\sqrt{3}:1$

Answer 1

The correct option is A $9:1$

Volume of 27 metal balls of radius r, $V_1=27\times \frac{4}{3}\pi r^3=36r^3\pi$

Volume of sphere of radius R, $V_2=\frac{4}{3}{R}^3\pi$

$\Rightarrow V_1=V_2$

$\Rightarrow 36r^3\pi =\frac{4}{3}{R}^3\pi \Rightarrow \frac{R^3}{r^3}=27$

$\Rightarrow R=3r$

Thus, $\frac{S_1}{S_2}=\frac{4\pi R^2}{4\pi r^2}=\frac{9}{1}$