Final answer:
The Moon's increasing distance from Earth will cause the length of a month to get longer due to the slower orbit as per Kepler's third law. Ultimately, this will lead to the day and the month being the same duration, approximately 47 current Earth days.
Step-by-step explanation:
As the Moon moves farther away from Earth, the radius of its orbit increases. Due to this change, using Kepler's third law, which relates the period of orbit to the radius, one can infer that the orbital period of the Moon will increase gradually. This phenomenon means the length of a month will become longer over time, as it takes the Moon more time to complete one revolution around the Earth.
Darwin's calculations, confirmed by the Apollo 11 mission's lunar mirrors, predict that billions of years from now, both the day and the month will be approximately 47 of our current days. This is because as the Moon orbits more slowly and moves away from Earth by about 3.8 centimeters per year, the Earth's rotation speed also slows down. Eventually, the Moon will reach a state where it is stationary in the sky over a particular spot on Earth.