Question

The diameter of a sphere is $6cm$. It is melted and drawn into a wire of diameter $0.2cm$. Find the length of the wire.

Answer 1

Step 1: Find the volume of the sphere.

Given that, the diameter of the sphere is $6cm$

Thus, the radius $R$ of the sphere $=\frac{6}{2}cm$ $(Radius=\frac{Diameter}{2})$

$=3cm$

The volume of the sphere of radius $R$ $=\frac{4}{3}\pi R^3$

$=\frac{4}{3}\mathrm{\pi }\times 3^3$

Step 2: Find the volume of the wire.

Given that, the diameter of the wire is $0.2cm$

Thus, the radius $r$ of the sphere $=\frac{0.2}{2}cm$ $(Radius=\frac{Diameter}{2})$

$=0.1cm$

Let the length of the wire be $hcm$

The volume of the cylinder of radius $r$ and length $h$ $=\pi r^{2}h$

The volume of the wire $=\mathrm{\pi }\times {\left(0.1\right)}^2\times \mathrm{h}$

Step 3: Find the length of the wire.

We have, the volume of the wire $=$volume of the sphere

$$\begin{gathered} \therefore \pi \times {\left(0.1\right)}^2\times h=\frac{4}{3}\pi \times 3^3 \\ \Rightarrow 0.01\times h=4\times 3^2 \\ \Rightarrow h=\frac{4\times 9}{0.01} \\ \Rightarrow h=3600cm \\ \Rightarrow h=36m(1m=100cm) \end{gathered}$$

Hence, the length of the wire is $36m$