- Identify the relationship: \(P = a^{\frac{3}{2}}\).
- Substitute the value \(P = 29.5\): \(29.5 = a^{\frac{3}{2}}\).
- Solve for \(a\) by calculating \(29.5^{\frac{2}{3}} \approx 9.547\).
- Select the matching option, resulting in \(\boxed{9.5 \text{ AU}}\).
Explanation
- Analyzing the Problem and Data
The problem asks for the distance \(a\) of Saturn from the sun in astronomical units (AU), given its orbital period \(P = 29.5\) years and the relationship \(P = a^{\frac{3}{2}}\). Our objective is to solve this equation for \(a\).
- Solving for the Distance 'a'
To isolate \(a\), we start with the formula \(P = a^{\frac{3}{2}}\). Substituting the known value \(P = 29.5\), we have the equation:
\[29.5 = a^{\frac{3}{2}}\]
To solve for \(a\), we raise both sides to the power of \(\frac{2}{3}\):
\[a = 29.5^{\frac{2}{3}}\]
Calculating this exponentiation:
\[a \approx 9.547\]
Comparing this result to the given options (9.5, 19.7, 44.3, 160.2), we find that 9.5 is the closest value.
- Conclusion
Based on the calculation, the distance from the sun for Saturn is approximately 9.5 AU. This matches the first option provided. Thus, the final answer is 9.5 AU.
Examples
Kepler's third law of planetary motion relates the orbital period of a planet to its distance from the sun. This is a fundamental concept in astronomy, allowing scientists to estimate the size of a planetary system by observing how long it takes planets to complete their orbits. By measuring the orbital period of an exoplanet, astronomers can use this same formula to determine its distance from its host star, helping us understand the layout and potential habitability of distant solar systems.