Answer :
To determine the orbital period of Jupiter using the stated equation \( P^2 = A^3 \), where \( A \) is the semi-major axis of Jupiter's orbit in astronomical units (AU) and \( P \) is the orbital period in years, follow these steps:
1. Identify the Given Value:
- The semi-major axis \( A \) for Jupiter is 5.2 AU.
2. Calculate \( A^3 \):
[tex]\[ A^3 = 5.2^3 = 140.608 \][/tex]
3. Set Up the Equation \( P^2 = A^3 \):
[tex]\[ P^2 = 140.608 \][/tex]
4. Solve for \( P \) by Taking the Square Root of Both Sides:
[tex]\[ P = \sqrt{140.608} \approx 11.8578 \][/tex]
5. Examine the Multiple-Choice Options:
- The options provided are 5 years, 11 years, 110 years, and 500 years.
6. Determine the Closest Match to the Calculated \( P \):
- The closest value to \( 11.8578 \) among the given choices is 11 years.
Therefore, the orbital period of Jupiter is approximately 11 years.
1. Identify the Given Value:
- The semi-major axis \( A \) for Jupiter is 5.2 AU.
2. Calculate \( A^3 \):
[tex]\[ A^3 = 5.2^3 = 140.608 \][/tex]
3. Set Up the Equation \( P^2 = A^3 \):
[tex]\[ P^2 = 140.608 \][/tex]
4. Solve for \( P \) by Taking the Square Root of Both Sides:
[tex]\[ P = \sqrt{140.608} \approx 11.8578 \][/tex]
5. Examine the Multiple-Choice Options:
- The options provided are 5 years, 11 years, 110 years, and 500 years.
6. Determine the Closest Match to the Calculated \( P \):
- The closest value to \( 11.8578 \) among the given choices is 11 years.
Therefore, the orbital period of Jupiter is approximately 11 years.
