https://file.aiquickdraw.com/mathbot/file/1726500393870nnuho1pt.jpg

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### Given Information:
- The formula for the orbital period \( P \) (in Earth years) is:
  \[
  P = \sqrt{d^3}
  \]
  where:
  - \( P \) is the orbital period of a planet in Earth years.
  - \( d \) is the average distance from the Sun in astronomical units (AU).
  
- Jupiter's distance from the Sun is given as \( d = 5.2 \, \text{AU} \).

### Part (a): Simplify the formula
The formula is already quite simple:
\[
P = \sqrt{d^3}
\]
No further simplification is required, as it's in its simplest form.

### Part (b): Calculate Jupiter's Orbital Period
We need to find the orbital period of Jupiter by substituting \( d = 5.2 \, \text{AU} \) into the formula.

\[
P = \sqrt{(5.2)^3}
\]
Let me calculate that for you.Jupiter's orbital period is approximately 11.86 Earth years.

### Summary:
- **(a)** The formula \( P = \sqrt{d^3} \) is already simplified.
- **(b)** Jupiter's orbital period is about **11.86 Earth years**.

Do you want more details on any part of the solution?

Here are five related questions you might find useful:
1. How does the formula \( P = \sqrt{d^3} \) relate to Kepler's third law of planetary motion?
2. What is the significance of the distance unit (AU) in astronomy?
3. How would the orbital period change if Jupiter were closer to the Sun?
4. How can this formula be modified to calculate orbital periods in units other than Earth years?
5. What is the difference between sidereal and synodic orbital periods?

**Tip**: Always check the units in the formula when solving astronomical problems, as they may change the final result.

a. Simplify the formula. b. What is Jupiter’s orbital period?

Learn how to simplify the orbital period formula P = sqrt(d^3) and calculate Jupiter’s orbital period, which is approximately 11.86 Earth years. Understand the mathematical concepts behind Kepler’s third law and astronomical units.

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