The moon is receding from the Earth by approximately 3.8 cm per year. Earth's mass is 5.98 x 10^24 kg, and its radius is 6.38 x 10^6 m. The Moon's mass is 7.3 x 10^22 kg, its radius is 1.79 x 10^6 m, and its orbital period around Earth is 27.3 days. The current average distance between the 2 surfaces is 3.85 x 10^8 m. Assume that neither body gains or loses mass and that the recession continues at a rate of 3.8 cm per year.

a.) Approximately how much will the gravitational attraction between the Moon and Earth change between now and 499 million years from now?
b.) Approximately how long, in present Earth-days, will it take the Moon to orbit Earth 499 million years from now?

To answer both parts of the question, we need to understand how the gravitational force between two objects is calculated. The gravitational force between two objects is given by the formula:

F = (G * m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant (approximately 6.67 x 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.

a.) To determine how much the gravitational attraction between the Moon and Earth will change, we need to calculate the initial gravitational force and compare it to the force at a future time.

Initial gravitational force:
F_initial = (G * m1 * m2) / r_initial^2

where m1 is Earth's mass (5.98 x 10^24 kg), m2 is the Moon's mass (7.3 x 10^22 kg), and r_initial is the initial distance between the two surfaces (3.85 x 10^8 m).

Future gravitational force:
To calculate the future gravitational force, we need to know the future distance (r_future) between the two surfaces. We can find this by adding the current distance (3.85 x 10^8 m) to the recession rate (3.8 cm/year) multiplied by the number of years (499 million years).

r_future = r_initial + (recession_rate * time_in_years)

Convert the recession rate from cm/year to meters/year:
Recession_rate = 3.8 cm/year * (1 m/100 cm) = 0.038 m/year

Convert the number of years from million years to years:
Time_in_years = 499 million years * 10^6 years/million years

Now, we can calculate the future gravitational force:

F_future = (G * m1 * m2) / r_future^2

To find the change in gravitational attraction between now and 499 million years from now:

Change_in_force = F_future - F_initial

b.) To determine how long it will take the Moon to orbit Earth 499 million years from now, we need to calculate the future orbital period.

The current orbital period of the Moon is 27.3 days. Assuming the orbital period remains constant, we need to find the number of orbits that occur in 499 million years.

Number_of_orbits = (Time_in_years * 365.25 days/year) / Orbital_period

Now, we can calculate the future orbital period:

Future_orbital_period = (Number_of_orbits * Orbital_period) / 365.25 days/year

Future_orbital_period will give us the answer in days, which is the final unit required in the question.

By following these steps and using the given values, you can find the answers to both parts of the question.