Answer :
Answer:
To determine Jupiter's farthest approach to Earth, we need to use the concept of the distance between two planets in their orbits around the Sun.
1. **Closest Approach (Perigee):** Given as \(6.28 \times 10^8\) km.
2. **Farthest Approach (Apogee):** This is when Jupiter is on the opposite side of the Sun from Earth. The distance between the two planets in this position is the sum of their distances from the Sun.
Let's denote:
- The average distance from Earth to the Sun (1 AU) as \( d_{\text{Earth}} \),
- The average distance from Jupiter to the Sun as \( d_{\text{Jupiter}} \),
- The closest approach as \( d_{\text{closest}} \),
- The farthest approach as \( d_{\text{farthest}} \).
**The Closest Approach:**
\[ d_{\text{closest}} = d_{\text{Jupiter}} - d_{\text{Earth}} \]
\[ 6.28 \times 10^8 \text{ km} = d_{\text{Jupiter}} - d_{\text{Earth}} \]
**The Farthest Approach:**
\[ d_{\text{farthest}} = d_{\text{Jupiter}} + d_{\text{Earth}} \]
To find \( d_{\text{Jupiter}} \) and \( d_{\text{Earth}} \), we need to use the distance between the Earth and Jupiter at their closest and farthest points.
3. **Average Distance of Jupiter from the Sun:**
The average distance from Jupiter to the Sun is about \( 7.78 \times 10^8 \) km (5.2 AU).
4. **Average Distance of Earth from the Sun:**
The average distance from Earth to the Sun is about \( 1.496 \times 10^8 \) km (1 AU).
Given:
\[ d_{\text{Jupiter}} = 7.78 \times 10^8 \text{ km} \]
\[ d_{\text{Earth}} = 1.496 \times 10^8 \text{ km} \]
Substituting into the equation for closest approach:
\[ 6.28 \times 10^8 = 7.78 \times 10^8 - 1.496 \times 10^8 \]
\[ 6.28 \times 10^8 = 6.28 \times 10^8 \text{ km} \]
**To find the farthest approach:**
\[ d_{\text{farthest}} = d_{\text{Jupiter}} + d_{\text{Earth}} \]
\[ d_{\text{farthest}} = 7.78 \times 10^8 + 1.496 \times 10^8 \]
\[ d_{\text{farthest}} = 9.276 \times 10^8 \text{ km} \]
Thus, Jupiter’s farthest approach to Earth is approximately \(9.28 \times 10^8\) km.Explanation:
