Question

A sphere of diameter 6.0 is moulded into a thin uniform wire of diameter 0.2mm.nCalculate the length of the wire in meters.(Take π=22/7)​

Answer 1

$\mathcal{\green{\underline{\blue{GIVEN:}}}}$

• The Diameter of sphere = 6.0 cm. Radius ⇒ ½(Diameter) ⇒ ½(6.0) ⇒ 3 cm.
• The Diameter of wire = 0.2 mm ⇒ 0.02 cm. [Convert the units so that all units are uniform]. And, Radius ⇒ (½)Diameter ⇒  (½)0.02 ⇒ 0.01 cm.

$\mathcal{\green{\underline{\blue{TO \:\: FIND:}}}}$

We need to calculate the length of the wire (l)

$\mathcal{\green{\underline{\blue{FORMULA \: \: USED:}}}}$

If we assume r₁ as length of sphere and r₂ as length of wire and height of the wire = h, then:

$\boxed{\bold{\large{volume \:\: of \:\: sphere=\dfrac{4}{3}\pi (r_1)^3}}}$

$\boxed{\bold{\large{volume \:\: of \:\: cylinder =\pi (r_2)^2h }}}$

We are considering the volume of the cylinder only because a wire is a small cylinder.

$\mathcal{\green{\underline{\blue{SOLUTION:}}}}$

The sphere is moulded into a wire. So they are not adding or removing the volume, they just changed the shape. In other words, the volume of the sphere is equal to the volume of wire.

$\sf{\red{\therefore Volume \:\: of \:\: cylinder = Volume \:\: of \:\: wire}}$

$\to \sf \dfrac{4}{3} \pi (r_1)^3=\pi( r_2)^2h$

Substitute the values.

$\to \sf \dfrac{4}{3} \not{\pi} (3)^3=\not{\pi} ( 0.01)^2h$

$\to \sf 36=0.0001 \times h$

$\to \sf h=\dfrac{36}{0.0001}$

$\to \sf h=360000cm.$

We need to calculate the length of wire in meters, so we need to convert cm into m. To convert cm into m, we need to divide the length value by 100

$\sf \to h=(\dfrac{360000}{100})m$

$\boxed{\sf{\blue{h=3600m}}}$

∴Length of wire = 3600 m.