Geometry

Solids

Medium

The diameter of a solid sphere is 6 cm. If the sphere is melted and drawn into a wire of diameter 2 mm, the length of the wire will be

12m

18m

24m

None of these

✅ Correct Option: 4

Sphere diameter = $6$ cm, so radius = $3$ cm

Wire diameter = $2$ mm = $0.2$ cm, so radius = $0.1$ cm

When a solid is melted and reshaped, its volume remains constant.

Volume of sphere = $\dfrac{4}{3}\pi r^3$

Volume = $\dfrac{4}{3} \times \pi \times (3)^3$

Volume = $\dfrac{4}{3} \times \pi \times 27$

Volume = $36\pi$ cm³

The wire is cylindrical in shape.

Volume of cylinder (wire) = $\pi r^2 h$ where $h$ = length of wire

Since volumes are equal:

$36\pi = \pi \times (0.1)^2 \times \text{length}$

$36\pi = \pi \times 0.01 \times \text{length}$

$36\pi = 0.01\pi \times \text{length}$

$36 = 0.01 \times \text{length}$

$\text{length} = \dfrac{36}{0.01}$

$\text{length} = 3600$ cm

Converting to meters: $3600$ cm = $36$ meters

Therefore, the length of the wire will be $3600$ cm or $36$ meters.

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Solution

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