Final answer:
The option that correctly shows acceleration due to gravity as inversely proportional to the square of the radius from a planet's center is B) a = k/r^2. This relationship is derived from Newton's law of universal gravitation.
Step-by-step explanation:
The correct option that shows that acceleration due to gravity is inversely proportional to the square of the radius from the center of the planet, like Earth, is B) a = k/r2. This formula represents how acceleration (a) changes with respect to the distance (r) from the center of the Earth (or any other spherically symmetric body).
In the context of gravity, Newton's law of universal gravitation states that the gravitational force (and thus acceleration due to gravity) between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The constant of proportionality (k) in the context of gravity is represented by the gravitational constant (G), multiplied by the mass of the Earth (M), thus the formula for the acceleration due to gravity is often seen as g = GM/r2, where G is the gravitational constant and M is the mass of the Earth.