Final answer:
If Earth had three times the radius but the same mass, an object's weight would decrease to 1/9 of its original weight due to the inverse square law of gravity. The weight is directly proportional to the mass of Earth and inversely proportional to the square of the distance from the center.
Step-by-step explanation:
Weight Change on an Altered Earth
If the Earth had three times the radius, but the same mass, an object's weight would change significantly because weight is directly proportional to the gravitational force exerted by Earth. The formula for gravitational force is:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between the centers of the two masses. Since the radius is three times larger, the distance r in the formula will increase by a factor of three. According to the inverse square law represented in the formula, the weight of an object would decrease by a factor of 1/3^2, or 1/9 of the original weight.
In the case where the Earth had only one-third its present mass, with the original size, the weight of a person would be reduced by the same factor of 1/3 since mass is decreased but the distance remains constant. Therefore, the gravitational force at Earth's surface would be one-third, and hence a person would weigh one-third as much.
Summarizing the rule, if you alter the mass of Earth, the weight changes proportionally to the mass, and if you alter the distance from the center of Earth (radius), the weight changes proportionally to the square of the change in distance.