Question

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

8.5 au
160.2 au
19.7 au
44.3 au

Explanation:

Step1: Start with the given formula

We have the formula \( P = a^{\frac{3}{2}} \), and we know that \( P = 29.5 \) years. We need to solve for \( a \).
First, rewrite the formula to solve for \( a \). Raise both sides to the power of \( \frac{2}{3} \) to isolate \( a \):
\( a = P^{\frac{2}{3}} \)

Step2: Substitute the value of \( P \)

Substitute \( P = 29.5 \) into the formula:
\( a = (29.5)^{\frac{2}{3}} \)

First, calculate \( 29.5^{\frac{1}{3}} \) (the cube root of 29.5) and then square the result.
The cube root of 29.5 is approximately \( \sqrt[3]{29.5} \approx 3.09 \) (using a calculator).
Then square this result: \( (3.09)^2 \approx 9.5 \)

Answer:

9.5 AU