One law of planetary motion states that the square of a planets orbital period, T, is proportional to the cube of the radius, r. This relationship can be written as an equation: T2=r3.

If the orbital period of a planet is 29.46 years. Calculate the distance the planet is away from the Sun (in Astronomical Units).
Responses

1.52 AU
1.52 AU - no response given

9.54 AU
9.54 AU - no response given

19.18 AU
19.18 AU - no response given

5.20 AU

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
17 days ago

To solve for the distance of the planet from the Sun in Astronomical Units (AU), we can use Kepler's Third Law of Planetary Motion, which states that

\[ T^2 \propto r^3 \]

We can express this as:

\[ T^2 = k \cdot r^3 \]

where \( k \) is a constant that can be determined based on Earth’s orbit, where \( T = 1 \) year and \( r = 1 \) AU. For Earth:

\[ 1^2 = k \cdot 1^3 \]
\[ k = 1 \]

Now we can use this relationship to determine \( r \) for the given orbital period \( T = 29.46 \) years:

1. Calculate \( T^2 \):

\[ T^2 = (29.46)^2 = 869.0516 \]

2. Set that equal to \( r^3 \):

\[ r^3 = 869.0516 \]

3. Now solve for \( r \) by taking the cube root:

\[ r = \sqrt[3]{869.0516} \]

Calculating the cube root:

\[ r \approx 9.54 \, \text{AU} \]

Thus, the planet is approximately 9.54 AU away from the Sun.

Therefore, the correct response is:

**9.54 AU**