Question

A man wants to make a small sphere of size of 1 cm of radius from a large sphere of size of 6 cm of radius. Find out how many such spheres can be made.

Answer 1

Hint: Assume the radius of the large sphere as ‘R’ and the radius of each small sphere as ‘r’. Now, assume that the total number of small spheres that can be formed is ‘n’. Use the volume relation given by: volume of large sphere is equal to ‘n’ times the volume of one small sphere. Mathematically, $\dfrac{4}{3}\pi {{R}^{3}}=n\times \dfrac{4}{3}\pi {{r}^{3}}$, as the volume of sphere is given by: $\dfrac{4}{3}\pi {{(radius)}^{3}}$.

Complete step-by-step solution -
Let us assume that the radius of the large sphere is ‘R’ and the radius of each small sphere is ‘r’. Also, let us assume that the number of small spheres that can be formed by the larger sphere is ‘n’.

Now, since the total volume larger sphere remains constant after ‘n’ smaller spheres are formed. Therefore,
$\begin{align}
  & \text{volume of large sphere}=n\times \left( \text{volume of 1 small sphere} \right) \\
 & \Rightarrow \dfrac{4}{3}\pi {{R}^{3}}=n\times \left( \dfrac{4}{3}\pi {{r}^{3}} \right) \\
\end{align}$
Cancelling the common terms, we get,
${{R}^{3}}=n\times {{r}^{3}}$
Now, substituting $R=6\text{ and r}=1$, we get,
$\begin{align}
  & {{6}^{3}}=n\times {{1}^{3}} \\
 & \Rightarrow n=216 \\
\end{align}$
Therefore, The total number of small spheres that can be formed from the large sphere is 216.

Note: One may note that we have used the volume relation to solve the question. This is because when the larger sphere is divided into smaller spheres then the total volume will remain the same. This is because there is no change in mass and density of the material. Never use area relation as it will be the wrong approach.