Question

The diameter of a metallic sphere Is $6cm$. The sphere is melted and drawn into a wire of uniform circular cross-section. If the length of the wire Is $36m$, Find the radius of its cross-section.

Answer 1

Step 1.Finding radius of the metallic sphere.

Diameter of metallic sphere $=6cm$

Therefore radius $\left(r\right)=\frac{6}{2}=3cm$

Step 2.Converting meter to cm.

The length of the wire $\left(h\right)=36m=3600cm\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{1}\mathbf{}\mathbf{m}\mathbf{}\mathbf{=}\mathbf{}\mathbf{100}\mathbf{}\mathbf{cm}\mathbf{}\mathbf{]}$

Step 3.Finding volumes

When an object is melted and recast into another object its volume remain constant.

So, we will use this concept to find the radius of the cross - section.

• Volume of metallic sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$

$=\frac{4}{3}\mathrm{\pi }{\left(3\right)}^3=36\mathrm{\pi }{\mathrm{cm}}^3$

• .Let the radius of wire of uniform circular cross-section be $Rcm$

Therefore its volume $=\pi R^2\mathrm{h}$

$$\begin{gathered} =\pi R^{2}x3600 \\ =3600{\mathrm{\pi R}}^2 \end{gathered}$$

Step 4. Finding the radius of the cross section

Since the Volume of metallic sphere$=$ volume of Wire Of Uniform Circular Cross-section

$$\begin{gathered} \Rightarrow 36\mathrm{\pi }=3600{\mathrm{\pi R}}^2 \\ \Rightarrow \frac{36\mathrm{\pi }}{3600\mathrm{\pi }}=R^2 \\ \Rightarrow \frac{1}{100}=R^2 \\ \Rightarrow R^2=\frac{1}{100} \\ \Rightarrow R^{}=\frac{1}{10}=0.1cm \end{gathered}$$

Hence the radius of the cross section is $0.1cm$