Question

The diameter of a metallic sphere is equal to \[9cm\]. It is melted and drawn into a long wire of diameter \[2mm\] having a uniform cross-section. Find the length of the wire.

Answer 1

Hint: When a solid is transformed into another shape the volume of both the solids remain equal. We can write it as, volume of sphere = volume of cylindrical wire, where \[\dfrac{4}{3}\pi {{r}^{3}}\] is volume of sphere and \[\pi {{\text{r}}^{2}}h\] is volume of cylindrical wire.

Complete step by step answer:
We have given that the diameter of the sphere is \[9cm\], that means the radius of the sphere is \[\dfrac{9}{2}cm\].
As we know, volume of sphere = \[\dfrac{4}{3}\pi {{r}^{3}}\]
We are putting the radius of sphere in the formula of vol. of sphere, we will get,
\[\Rightarrow \text{vol}\text{. of sphere =}\dfrac{4}{3}\pi {{\left( \dfrac{9}{2} \right)}^{3}}\]
\[\Rightarrow \text{vol}\text{. of sphere =}\dfrac{243}{2}\pi ......\left( i \right)\]
Assume, the length of the cylindrical wire be ‘l’
And we know that, volume of cylindrical wire = \[\pi {{\text{r}}^{2}}h\], where ‘r’ is the radius of cross section and ‘h’ is the length of wire.
We are given the diameter of the long wire which is equal to \[2mm\], so we will get a radius of cylindrical wire as \[1mm=\dfrac{1}{10}cm=0.1cm\].
Now, here we are putting the values of ‘h’ and ‘r’ in vol. of cylindrical wire, we may get,
\[\Rightarrow \text{vol}\text{. of cylindrical wire =}\pi {{\left( 0.1 \right)}^{2}}l\]
\[\Rightarrow \text{vol}\text{. of cylindrical wire =(0}\text{.01)}\pi l......\left( ii \right)\]
As we know that when a solid is transformed into another shape volume remains constant, that means, Volume of sphere = volume of cylindrical wire……(iii)

From (i), (ii) and (iii), we will get,
\[\dfrac{243}{2}\pi =\text{(0}\text{.01)}\pi l\]
\[\Rightarrow l=\dfrac{\left( 243 \right)}{2\left( 0.01 \right)}\]
\[\Rightarrow l=12150cm\]

Therefore, we can say that the length of the cylindrical wire of diameter \[2mm\] is \[12150cm\].

Note: When a solid body is transformed into another shape the volume of both the bodies remains equal. In a hurry, students usually make mistakes in the writing formula of volume of sphere and volume of cylinder. Also, the possible mistake one can make is not changing the units of either radius of cylinder or the units of radius of sphere. Also, remember that we are given with diameter and in formulas we need radius, so make sure you find radius before putting the values.