Question

The radius of a solid iron sphere is 8cm. Eight rings of iron plate of external radius \[6\dfrac{2}{3}\ cm\] and thickness 3cm are made by melting this sphere. Find the internal diameter of each ring.

Answer 1

Hint: As the new rings are made up of the material which we got after melting the solid iron sphere, hence, the total volume of the solid iron sphere is equal to the sum of the volume of all the 8 rings together.
The formula to calculate the volume of a solid sphere is equal to
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
Now, the rings are equivalent to hollow cylinders whose volume can be calculated as follows
\[=\pi \left( {{r}_{2}}^{2}-{{r}_{1}}^{2} \right)h\]
(Where \[{{r}_{2}}\] is the outer radius and \[{{r}_{1}}\] is the inner radius of the hollow cylinder)

Complete step-by-step answer:
As mentioned in the question, we have to find the internal diameter of the hollow cylinder.
Now, the volume of the solid iron sphere is
\[\begin{align}
  & =\dfrac{4}{3}\pi {{r}^{3}} \\
 & =\dfrac{4}{3}\pi {{8}^{3}} \\
 & =\dfrac{4}{3}\pi 8\times 8\times 8\ c{{m}^{3}} \\
\end{align}\]
Now, this sphere is to be melted and from the molten material 8 rings are to be made which has 3 cm and outer radius \[6\dfrac{2}{3}\ cm\], hence, by using the formula given in the hint, we get that the volume of a single ring is
\[\begin{align}
  & =\pi \left( {{\dfrac{20}{3}}^{2}}-{{r}_{1}}^{2} \right)3 \\
 & =3\pi \left( \dfrac{400}{9}-{{r}_{1}}^{2} \right) \\
\end{align}\]
Now, the volume of 8 such rings is
\[\begin{align}
  & =8\times 3\pi \left( \dfrac{400}{9}-{{r}_{1}}^{2} \right) \\
 & =24\pi \left( \dfrac{400}{9}-{{r}_{1}}^{2} \right) \\
\end{align}\]
As mentioned in the hint, we can get that these two volumes can be equated as follows

\[\begin{align}
  & 24\pi \left( \dfrac{400}{9}-{{r}_{1}}^{2} \right)=\dfrac{4}{3}\pi 8\times 8\times 8 \\
 & \left( \dfrac{400}{9}-{{r}_{1}}^{2} \right)=\dfrac{64\times 4}{9} \\
 & {{r}_{1}}^{2}=\dfrac{144}{9} \\
 & {{r}_{1}}=\dfrac{12}{3}\ cm \\
 & {{r}_{1}}=4\ cm \\
\end{align}\]
Hence, the internal radius is 4 cm and so the internal diameter is 8 cm.

NOTE: -
The students can make an error if they don’t know the fact that as the new rings are made up of the material which we got after melting the solid iron sphere, hence, the total volume of the solid iron sphere is equal to the sum of the volume of all the 8 rings together.
Also, the students cannot get to the right answer for this question if they don’t know the important formulae that are mentioned in the hint as follows
The formula to calculate the volume of a solid sphere is equal to
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
Now, the rings are equivalent to hollow cylinders whose volume can be calculated as follows
\[=\pi \left( {{r}_{2}}^{2}-{{r}_{1}}^{2} \right)h\]
(Where \[{{r}_{2}}\] is the outer radius and \[{{r}_{1}}\] is the inner radius of the hollow cylinder)