Question

Twenty-seven solid iron spheres, each of radius \[r\] and surface area \[S\] are melted to form a sphere with surface area \[S'\]. Find the ratio of \[S{\text{ and }}S'\].

Answer 1

Hint: First of all, consider the radius of the new sphere as a variable and find the volumes of the iron sphere and the new sphere. Then Equate their volume by the given condition to find the ratio of radii of the iron sphere and new sphere which will lead you to solve the solution of the given problem.

Complete Step-by-step answer:
Given radius of the iron sphere = \[r\]
Surface area of each iron sphere = \[S\]
Surface area of new sphere = \[S'\]
Let the radius of new sphere = \[R\]

We that the volume of a sphere with of radius \[r\] is given by \[V = \dfrac{4}{3}\pi {r^3}\]
So, the area of each iron sphere \[ = \dfrac{4}{3}\pi {r^3}\]
And the area of new sphere \[ = \dfrac{4}{3}\pi {R^3}\]
We know that whenever one shape is melted to another shape the volume remains constant.
As 27 iron spheres are melted to a new sphere, we have
Volume of 27 iron sphere = Volume of new sphere
\[
  27 \times \dfrac{4}{3}\pi {r^3} = \dfrac{4}{3}\pi {R^3} \\
  27{r^3} = {R^3} \\
  \dfrac{{{r^3}}}{{{R^3}}} = \dfrac{1}{{27}} \\
  {\left( {\dfrac{r}{R}} \right)^3} = {\left( {\dfrac{1}{3}} \right)^3} \\
  \therefore \dfrac{r}{R} = \dfrac{1}{3}...............................................................\left( 1 \right) \\
\]
We know that the surface area of a sphere with radius \[r\] is given by \[4\pi {r^2}\]
So, the surface area of the iron sphere \[S = 4\pi {r^2}\]
And the surface area of the new sphere \[S' = 4\pi {R^2}\]
We have to find the ratio of \[S{\text{ and }}S'\], which is given by
\[
  \dfrac{S}{{S'}} = \dfrac{{4\pi {r^2}}}{{4\pi {R^2}}} \\
  \dfrac{S}{{S'}} = \dfrac{{{r^2}}}{{{R^2}}} \\
  \dfrac{S}{{S'}} = {\left( {\dfrac{r}{R}} \right)^2} \\
  \dfrac{S}{{S'}} = {\left( {\dfrac{1}{3}} \right)^2}{\text{ }}\left[ {{\text{Using }}\left( 1 \right)} \right] \\
  \therefore \dfrac{S}{{S'}} = \dfrac{1}{9} \\
\]
Thus, the ratio of \[S{\text{ and }}S'\] is \[1:9\].

Note: Whenever one shape is melted to another shape the volume remains constant. Surface area of a sphere with radius \[r\] is given by \[4\pi {r^2}\]. Volume of a sphere with of radius \[r\] is given by \[V = \dfrac{4}{3}\pi {r^3}\]