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# If the distance between Earth and the sun were half its present value, the number of days in a year would be

$(a)\;64.5 \quad (b)\;129 \quad (c)\;182.5 \quad (d)\;739$

By Kepler's law
$T^2 \;\alpha \; a^3$
$\large\frac{T_1^2}{a_1 ^3}=\frac{T_2^2}{a_2^3}$
$T_2^2 =\bigg( \large\frac{a_2}{a_1}\bigg)^3 $$\times T_1^2 T_2^2 =\bigg( \large\frac{1}{2}\bigg)^3$$ \times T_1^2$
$T_2^2 =\bigg( \large\frac{1}{8}\bigg) $$\times 1\;year T_2 =\large\frac{1}{\sqrt 8}$$ \times 365\;days$
$\quad=129\;days$
Hence b is the correct answer.

edited Feb 17, 2014 by meena.p