Question

the orbital period, \\(p\\), of a planet and the planets distance from the sun, \\(a\\), in astronomical units is related by the formula \\(p = a^{\frac{3}{2}}\\). if saturns orbital period is 29.5 years, what is its distance from the sun?

9.5 au
19.7 au
44.3 au
160.2 au

Explanation:

Set up the equation

Using the Keplers Third Law knowledge point

$$ P = a^{\frac{3}{2}} $$

$$ 29.5 = a^{\frac{3}{2}} $$

Solve for the distance

Using the Rational Exponents knowledge point

$$ a = (29.5)^{\frac{2}{3}} $$

$$ a \approx 9.547\text{ AU} $$

Match with the options

The calculated value \(a \approx 9.5\text{ AU}\) matches the first option.

Answer:

• (A) 9.5 AU (Correct answer)
• (B) 19.7 AU
• (C) 44.3 AU
• (D) 160.2 AU