question_answer

A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is \[\text{1}.\text{5}\times \text{1}{{0}^{\text{8}}}\text{ km}\] away from the sun?

Answer:

               

Earth	Saturn
\[{{T}_{1}}=T\]	\[{{T}_{2}}=29.5\,T\]
\[{{R}_{1}}=1.5\times {{10}^{8}}km\]	\[{{R}_{2}}=?\]

We know \[\frac{{{T}^{2}}}{{{R}^{3}}}=const\] \[\Rightarrow {{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{3}}={{\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)}^{2}}\] \[\frac{{{R}_{2}}}{{{R}_{1}}}={{\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)}^{\frac{2}{3}}}\] On substitution, we get \[{{R}_{2}}=1.5\times {{10}^{8}}\times {{\left( \frac{29.5}{1} \right)}^{\frac{2}{3}}}\]                 \[=14.32\times {{10}^{11}}km\]