Melting of Sphere into a Cylindrical Wire

Correct Answer: None of the above

The correct answer is Option E.

Method 1: Basic Method

Step 1

The diameter of the sphere is 6 cm.

Therefore, its radius is:

\[ R=\frac{6}{2}=3 \]

Step 2

Volume of the sphere:

\[ V=\frac{4}{3}\pi R^3 \] \[ =\frac{4}{3}\pi(3)^3 \] \[ =36\pi \]

Step 3

The melted sphere is converted into a wire of length 36 cm.

The wire is cylindrical in shape.

Volume of the wire:

\[ V=\pi r^2h \] \[ =\pi r^2(36) \]

Step 4

Since volume remains unchanged:

\[ 36\pi=36\pi r^2 \] \[ r^2=1 \] \[ r=1\text{ cm} \]

But 1 cm is not given in Options A, B or C.

Correct Answer: Option E

Method 2: Competitive Exam Short Method

Step 1

Sphere radius:

\[ R=3 \]

Step 2

Volume of sphere:

\[ \frac{4}{3}\pi(3)^3=36\pi \]

Step 3

Wire volume:

\[ \pi r^2\times36 \]

Step 4

\[ 36\pi=36\pi r^2 \] \[ r=1\text{ cm} \]

Answer: Option E

Method 3: Pen-Free / Mental Method

Step 1

The sphere has radius 3 cm.

Step 2

Its volume is:

\[ 36\pi \]

Step 3

The wire length is also 36 cm.

Therefore:

\[ 36\pi r^2=36\pi \] \[ r=1 \]

Since 1 cm is not available in the options:

Correct Answer: Option E

RRB Group D Science Set 2

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