Mathematics

By applying force on an uniform metallic wire, the wire is extended four times along its length with same width all around. Then :
(i) The volume of metal in both the cases is same.
(ii) The surface area of solid wires in both the cases is same.
Which of the above statements is/are true?
only (i)
only (ii)
both (i) and (ii)
neither (i) nor (ii)

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On transforming a solid from one shape to another by any external force, the volume of solid remains same.

Let initial length of wire be l and radius be r.

After extension,

Length of wire = 4l and Radius be R.

Since, volume is equal in both the cases :

∴ πr²h = πR²H

⇒ πr²l = πR².4l

⇒ r² = 4R²

⇒ r = $\sqrt{4R^2}$

⇒ r = 2R

⇒ R = $\dfrac{r}{2}$

Surface area in first case :

⇒ 2πrh

⇒ 2πrl

Surface area in second case :

⇒ 2πRh

⇒ $2π \times \dfrac{r}{2} \times 4l$

⇒ 4πrl.

∴ Surface area of solid wires in both the cases are different.

Hence, Option 1 is the correct option.

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