A conical toy tent-house, 28 cm in radius and 21 cm in height, is made from a rectangular sheet of paper 22 cm wide. The smallest length of the paper sheet required is :

1. 70 cm

2. 105 cm

3. 140 cm

4. 280 cm

Since, conical toy is made from the rectangular sheet.

∴ Surface area of cone = Area of rectangular sheet.

Let smallest length of the paper sheet required be a cm.

∴ πrl = length × breadth

$\Rightarrow πr\sqrt{r^2 + h^2} = \text{length × breadth} \\[1em] \Rightarrow \dfrac{22}{7} \times 28 \times \sqrt{28^2 + 21^2} = a \times 22 \\[1em] \Rightarrow 22 \times 4 \times \sqrt{784 + 441} = a \times 22 \\[1em] \Rightarrow 88 \times \sqrt{1225} = a \times 22 \\[1em] \Rightarrow 88 \times 35 = a \times 22 \\[1em] \Rightarrow a = \dfrac{88 \times 35}{22} \\[1em] \Rightarrow a = 4 \times 35 = 140 \text{ cm}.$

Hence, Option 3 is the correct option.