Question

A spherical ball of lead has been melted and made into identical smaller balls with radius equal to half the radius of the original one. How many such balls can be made?

Answer 1

Hint: The volume of original ball =$N \times$(volume of smaller ball), where N is the number of smaller balls made out of the given ball and the volume of spherical ball is given by $V = \dfrac{4}{3}\pi {r^3}$,where r is radius of spherical ball.

Complete step-by-step answer:
Given a spherical ball of lead has been melted and made into identical smaller balls with radius equal to half the radius of the original one.
To find – No. of balls that can be made.
Let the radius of the spherical ball be r.
Therefore, the volume of original ball $V = \dfrac{4}{3}\pi {r^3}$
According to the question, the radius of smaller balls is equal to half the radius of the original one.
Radius of smaller ball$= \dfrac{r}{2}$
Therefore, the Volume of smaller balls ${V_2} = \dfrac{4}{3}\pi {\left( {\dfrac{r}{2}} \right)^3} = \dfrac{4}{3}\pi \dfrac{{{r^3}}}{8} = \dfrac{{\pi {r^3}}}{6}$
Now, volume of original ball =$N \times$(volume of smaller ball), where N is the number of smaller
Therefore, the no. of smaller balls made out of the given ball –
$V = N \times {V_2} \\\ \Rightarrow N = \dfrac{V}{{{V_2}}} = \dfrac{{\dfrac{4}{3}\pi {r^3}}}{{\pi \dfrac{{{r^3}}}{6}}} = \dfrac{4}{3} \times 6 = 8 \\\$
So, 8 balls are made out of the given ball.

Note: Whenever this type of question appears, always proceed by equating the volume of the original ball with the total volume of the smaller balls, as the volume of the original ball remains the same. So, the no. of smaller balls made out of the given ball can be found out by using the equation volume of original ball =$N \times$(volume of smaller ball), where N is the number of smaller balls.