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?

44.3 au
19.7 au
9.5 au
160.2 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 \( a \).

Step2: Isolate \( a \)

First, we can rewrite the formula to solve for \( a \). Raise both sides of the equation to the power of \( \frac{2}{3} \) to isolate \( a \). So, \( a = P^{\frac{2}{3}} \).

Step3: Substitute the value of \( P \)

Substitute \( P = 29.5 \) into the formula for \( a \). So, \( a=(29.5)^{\frac{2}{3}} \).
First, calculate \( 29.5^{\frac{1}{3}} \approx 3.09 \) (using a calculator to find the cube root of 29.5). Then square that result: \( (3.09)^2 \approx 9.5 \).

Answer:

9.5 AU