If the radius of the Earth somehow bloated to twice its size, with no changes in mass, your weight at the Earth's new surface would be ___A) one-quarter.B) one eighth.C) the same.D) twice.E) half.

Half

To determine the effect of Earth's radius being doubled with no change in mass on your weight at the Earth's new surface, we need to consider the relationship between gravity and distance.

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 two objects
r is the distance between the centers of the two objects

In this case, we can assume that your weight is due to the gravitational force exerted by Earth. Therefore, m1 would be your mass, m2 would be the mass of Earth, and r would be the distance from the center of Earth to its surface.

Now, if the radius of Earth is doubled, the new distance from the center of Earth to its surface would be 2 times the original distance. Let's call the original radius "r0" and the new radius "r1". Therefore, r1 = 2 * r0.

Now, let's substitute the new values into the gravitational force formula:

F1 = (G * m1 * m2) / (r1^2)
F0 = (G * m1 * m2) / (r0^2)

To find the ratio of the new force (F1) to the original force (F0), we can divide the two equations:

(F1 / F0) = [(G * m1 * m2) / (r1^2)] / [(G * m1 * m2) / (r0^2)]
(F1 / F0) = [(G * m1 * m2) / (r1^2)] * [(r0^2) / (G * m1 * m2)]
(F1 / F0) = (r0^2) / (r1^2)
(F1 / F0) = (r0^2) / [(2 * r0)^2]

Simplifying further:

(F1 / F0) = (r0^2) / (4 * r0^2)
(F1 / F0) = 1 / 4

Based on this calculation, the ratio of the new force (F1) to the original force (F0) is 1/4.

Since weight is essentially the force experienced due to gravity, we can conclude that if the radius of Earth is doubled, your weight at the Earth's new surface would be one-quarter (one-fourth) of your weight at the original surface. Therefore, the correct answer is A) one-quarter.