The distance between Sun and Earth is $$R$$. The duration of year if the distance between Sun and Earth becomes $$3R$$ will be :

We use Kepler’s Third Law, which states that the square of the orbital period is proportional to the cube of the semi-major axis (orbital radius): $$T^2 \propto R^3$$.

For the original distance $$R$$ with period $$T_1 = 1$$ year: $$T_1^2 \propto R^3$$. For the new distance $$3R$$ with period $$T_2$$: $$T_2^2 \propto (3R)^3$$.

Taking the ratio gives $$\frac{T_2^2}{T_1^2} = \frac{(3R)^3}{R^3} = 27$$, so $$T_2^2 = 27 \times T_1^2 = 27$$ and hence $$T_2 = \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \text{ years}$$.

The duration of the year when the distance becomes $$3R$$ is $$3\sqrt{3}$$ years. The correct answer is Option B.