Calculate the distance, in m, of the centre of mass of the Earth-Moon system from the centre of the Earth. Note: the mass of the Moon is 7.35x10 22kg, the mass of the Earth is 5.98x10 24kg, and their separation is 3.85x10 8m.

To calculate the distance of the center of mass of the Earth-Moon system from the center of the Earth, we can use the concept of the center of mass formula.

The center of mass formula states that the center of mass of a system is given by the weighted average of the positions of the individual masses, where the weights are the masses of the objects.

In this case, we have two masses: the mass of the Moon (m1) and the mass of the Earth (m2).
The distance between their centers is given as the separation between the Earth and the Moon (r).

Let's denote the distance of the center of mass from the center of the Earth as x. Since the Earth has a much larger mass compared to the Moon, we can assume that the center of mass lies close to the center of the Earth.

We can set up the equation as follows:

(m1 * x1 + m2 * x2) / (m1 + m2) = x

where x1 is the distance from the center of the Moon to the center of mass, and x2 is the distance from the center of the Earth to the center of mass.

Since the center of mass is assumed to be close to the center of the Earth, we can approximate x1 to be equal to the separation r.

Substituting the values into the equation:

[(7.35x10^22 kg) * (3.85x10^8 m) + (5.98x10^24 kg) * x2] / [(7.35x10^22 kg) + (5.98x10^24 kg)] = x

Now we can solve for x by rearranging the equation:

(7.35x10^22 kg * 3.85x10^8 m + 5.98x10^24 kg * x2) / (7.35x10^22 kg + 5.98x10^24 kg) = x

By plugging in the given values, we can find the distance of the center of mass from the center of the Earth in meters.