Answer:
the average distance from the Sun of a planet with an orbital period of 45.66 years is approximately 12.708 astronomical units (AU).
Step-by-step explanation:
You can calculate the average distance from the Sun (in astronomical units, AU) of a planet using Kepler's Third Law of Planetary Motion, which relates the planet's orbital period (T) to its average distance from the Sun (a). The formula is:
a^3/T^2 = k
Where:
a is the average distance from the Sun in AU.
T is the orbital period in years.
k is a constant.
You need to find "a," so you can rearrange the formula:
a^3 = k * T^2
Now, plug in the values:
a^3 = k * (45.66 years)^2
To solve for "a," take the cube root of both sides:
a = (k * (45.66 years)^2)^(1/3)
You'll need to know the value of the constant "k," which relates to the properties of the Sun and is approximately equal to 1.000041 AU^3/year^2. Now, calculate "a":
a = (1.000041 AU^3/year^2 * (45.66 years)^2)^(1/3)
a ≈ (1.000041 AU^3/year^2 * 2080.9956 years^2)^(1/3)
a ≈ (2081.005 AU^3)^(1/3)
a ≈ 12.708 AU