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If the distance between the earth and the sun were half of its present value, the number of days in a year would have been…………..

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answer is 129.

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Detailed Solution

$T^{2} α r^{3} {(\frac{T_{1}}{T_{2}})}^{2} = {(\frac{r_{1}}{r_{2}})}^{3} \Rightarrow \frac{T_{1}}{T_{2}} = {(\frac{r_{1}}{r_{2}})}^{\frac{3}{2}}$

$T_{1} = 365, r_{1} = 2 r_{2} \Rightarrow r_{2} = \frac{r_{1}}{2}$

$\frac{T_{1}}{T_{2}} = {(\frac{r_{1}}{r_{2}})}^{\frac{3}{2}} \Rightarrow \frac{T_{1}}{T_{2}} = 2 \sqrt{2}$

$T_{2} = \frac{T_{1}}{2 \sqrt{2}} = \frac{365}{2 \sqrt{2}} = 129$

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