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 $\dfrac{1}{4}th$ 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

Hint: In order to solve this problem we need to calculate the surface area of a larger ball and then divide it by the total surface area of smaller balls but first we need to calculate the total number of smaller balls.

Complete step-by-step answer:

Let the radius of the smaller ball be r.
Given: Radius of each smaller ball = $\dfrac{1}{4}$ radius of original ball.
 r = $\dfrac{1}{4} \times $ Radius of original ball.
Radius of original ball = 4r.
Volume of original ball
$
  {V_1} = \dfrac{4}{3}\pi {r^3} \\
   = \dfrac{4}{3}\pi {(4r)^3} \\
$
Volume of smaller ball
$
  {V_2} = \dfrac{4}{3}\pi {r^3} \\
   = \dfrac{4}{3}\pi {(r)^3} \\
 $
Number of balls is given by $ = \dfrac{{{\text{ Volume of original ball}}}}{{{\text{Volume of smaller ball}}}}$
$ = \dfrac{{\dfrac{4}{3}\pi {{(4r)}^3}}}{{\dfrac{4}{3}\pi {{(r)}^3}}} = \dfrac{{64{r^3}}}{{{r^3}}} = 64$
Number of balls = 64
Now, Surface area of sphere $ = 4\pi {r^2}$
Surface area of original ball ${S_1} = 4\pi {(4r)^2}$
Surface area of smaller ball ${S_2} = 4\pi {(r)^2}$
Total surface area of 64 small balls, \[{S_3} = 64 \times 4\pi {(r)^2}\]
Now comparing the surface area of smaller balls with original ball
\[
   \Rightarrow \dfrac{{{S_3}}}{{{S_1}}} = \dfrac{{64 \times 4\pi {r^2}}}{{4\pi \times 16{r^2}}} \\
   \Rightarrow \dfrac{{{S_3}}}{{{S_1}}} = \dfrac{{64}}{{16}} \\
   \Rightarrow \dfrac{{{S_3}}}{{{S_1}}} = 4 \\
 \]
Hence, the total surface area of small balls is equal to 4 times the surface area of the original ball.

Note: To solve this type of question remember the formula of surface area and volume of all shapes. In this question the original ball is casted into smaller balls and radius is given in the form of condition similarly in other questions they may convert square into sphere and ask comparison of their surface areas. The approach remains to solve the same as we followed in this question.