Final answer:
The distance from the center of the Earth where the gravitational force will be zero is D/82.
Step-by-step explanation:
To find the distance from the center of the Earth where the gravitational force will be zero, we need to consider the gravitational forces exerted by both the Earth and the Moon. The gravitational force between two objects is given by the formula F = G * (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Since the mass of the Earth is 81 times the mass of the Moon, the force exerted by the Earth on an object will be 81 times greater than the force exerted by the Moon on the same object at the same distance.
Therefore, the gravitational force will be zero where the force exerted by the Moon is equal in magnitude but opposite in direction to the force exerted by the Earth. This can be represented by the equation 81 * (G * (m1 * m2) / r^2) = G * (m1 * m2) / (D-r)^2, where D is the distance between the centers of the Earth and the Moon.
Simplifying this equation, we can find the value of r where the gravitational force is zero. Solving for r, we get r = D/82. Therefore, the correct answer is A. D/82.