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The problem in the image is asking for an estimate of the time \( T \) for a new planet to travel around the Sun using **Kepler's Third Law**. Kepler's Third Law states that the square of the period of orbit (\( T \)) of a planet is proportional to the cube of the semi-major axis of its orbit (\( a \)):

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

In the form of an equation, it is usually written as:

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

where \( k \) is a constant, and \( a \) is the semi-major axis (mean distance from the Sun). When using astronomical units (AU) and Earth years, \( k = 1 \), and the equation simplifies to:

\[
T^2 = a^3
\]

### Given Information:
- The mean distance of the planet is **4004 million miles**.
- 1 Astronomical Unit (AU) is approximately **93 million miles**.

### Step-by-Step Solution:

1. **Convert the given distance to AU**:
   \[
   a = \frac{4004 \ \text{million miles}}{93 \ \text{million miles per AU}} = \frac{4004}{93} \approx 43.06 \ \text{AU}
   \]

2. **Apply Kepler's Third Law**:
   \[
   T^2 = (43.06)^3
   \]
   First, calculate the cube of 43.06:
   \[
   (43.06)^3 = 79745.7
   \]
   Now, take the square root to find \( T \):
   \[
   T = \sqrt{79745.7} \approx 282.48 \ \text{years}
   \]

### Final Estimate:
The planet would take approximately **282.48 Earth years** to complete one orbit around the Sun.

Let me know if you'd like further details or clarifications!

---

### Related Questions:
1. How does Kepler's Third Law change for non-circular orbits?
2. What factors could influence the actual time a planet takes to orbit the Sun?
3. Can we use Kepler's laws for objects other than planets, like comets or asteroids?
4. What are the limitations of using Kepler's Third Law for distant objects?
5. How would you estimate the orbital time of other celestial objects at similar distances?

### Tip:
Always remember to convert units (like miles to AU) when working with Kepler's laws to maintain consistency and accuracy!

A new planet is discovered beyond Pluto at a mean distance to the sun of 4004 million miles. Using Kepler's third law, determine an estimate for the time T to travel around the sun in an orbit.

Discover how to estimate the orbital period of a new planet beyond Pluto using Kepler's Third Law. Given the mean distance of 4004 million miles from the sun, the step-by-step solution shows how to calculate the time in Earth years for the planet to complete its orbit. This is suitable for high school or introductory college astronomy students.

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