To determine Saturn's distance from the sun, denoted as [tex]\( a \)[/tex], given its orbital period [tex]\( P \)[/tex] of 29.5 years, we can use the relation:
[tex]\[ P = a^{\frac{3}{2}} \][/tex]
We need to solve for [tex]\( a \)[/tex] when [tex]\( P = 29.5 \)[/tex]. Here are the steps to find [tex]\( a \)[/tex]:
1. Formula Setup: We start with the equation:
[tex]\[
29.5 = a^{\frac{3}{2}}
\][/tex]
2. Isolate [tex]\( a \)[/tex]: To isolate [tex]\( a \)[/tex], we need to undo the exponent [tex]\(\frac{3}{2}\)[/tex]. To do this, we raise both sides of the equation to the power of [tex]\(\frac{2}{3}\)[/tex]. This is because:
[tex]\[
\left(a^{\frac{3}{2}}\right)^{\frac{2}{3}} = a^{\left(\frac{3}{2} \cdot \frac{2}{3}\right)} = a
\][/tex]
So, applying this to both sides of the equation:
[tex]\[
\left(29.5\right)^{\frac{2}{3}} = a
\][/tex]
3. Calculate [tex]\( a \)[/tex]: Given the true result, [tex]\(\left(29.5\right)^{\frac{2}{3}}\)[/tex] is approximately equal to 9.547. Thus:
[tex]\[
a \approx 9.547
\][/tex]
4. Round or approximate value: Generally, distances in astronomy are given to a few significant digits. Therefore, rounding 9.547 to the nearest value among the choices provided, we get approximately:
[tex]\[
a \approx 9.5 \text{ AU}
\][/tex]
Therefore, Saturn's distance from the sun is approximately 9.5 AU.
The correct option is:
- [tex]\( \boxed{9.5 \text{ AU}} \)[/tex]
