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

Let the wire's radius be a.

Given, sphere is melted into the wire.

The wire formed is a cylinder, hence the volume of wire will be equal to the volume of sphere.

Radius of sphere (r) = $\dfrac{\text{Diameter}}{2}$

= $\dfrac{6}{2}$ = 3 cm.

Volume of sphere (V) = $\dfrac{4}{3}πr^3$

Putting values we get,

$V = \dfrac{4}{3}π \times (3)^3 \\[1em] = 4π \times 3^2 \\[1em] = 36π \text{ cm}^3.$

Given, length of wire = 36 m.

So, height of cylinder = 36 m = 3600 cm.

Volume of cylinder = V = 36π cm³.

$\therefore πr^2h = 36π \\[1em] \Rightarrow r^2 = \dfrac{36π}{πh} \\[1em] \Rightarrow r^2 = \dfrac{36}{3600} \\[1em] \Rightarrow r^2 = \dfrac{1}{100} \\[1em] \Rightarrow r = \sqrt{\dfrac{1}{100}} \\[1em] \Rightarrow r = \dfrac{1}{10} \text{ cm} = 1 \text{ mm}.$

Hence, the radius of the wire is 1 mm.