To find the distance of Saturn from the sun, we need to use the given relationship between the orbital period, [tex]\(P\)[/tex], and the distance from the sun, [tex]\(a\)[/tex], in astronomical units (AU). The relationship is given by the formula:
[tex]\[ P = a^{\frac{3}{2}} \][/tex]
Here, we are given that the orbital period of Saturn is [tex]\(P = 29.5\)[/tex] years. We need to find the distance [tex]\(a\)[/tex] in AU.
First, we start with the equation:
[tex]\[ P = a^{\frac{3}{2}} \][/tex]
We need to solve for [tex]\(a\)[/tex]:
[tex]\[ a^{\frac{3}{2}} = P \][/tex]
Taking the power of [tex]\(\frac{2}{3}\)[/tex] on both sides to solve for [tex]\(a\)[/tex]:
[tex]\[ a = P^{\frac{2}{3}} \][/tex]
Substitute the given orbital period [tex]\(P = 29.5\)[/tex] years into the equation:
[tex]\[ a = 29.5^{\frac{2}{3}} \][/tex]
By evaluating the expression [tex]\(29.5^{\frac{2}{3}}\)[/tex], we get:
[tex]\[ a \approx 9.547 \][/tex]
Thus, Saturn's distance from the sun is approximately [tex]\(9.547\)[/tex] AU. Given the choices:
- 9.5 AU
- 19.7 AU
- 44.3 AU
- 160.2 AU
The closest choice to our calculated value is:
[tex]\[ \boxed{9.5 \text{ AU}} \][/tex]
