How many kilometers would you have to go above the surface of the earth for your weight to decrease to 39{\rm \\%} of what it was at the surface?

Express your answer using two significant figures.

G M/r^2 = .39 G M/Re^2

r^2 = Re^2/.39

To solve this problem, we need to understand the concept of gravitational force and how it relates to the distance from the surface of the Earth. The force of gravity on an object can be expressed as:

F = (G * m * M) / r^2

Where:
F is the force of gravity,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3/kg/s^2),
m is the mass of the object,
M is the mass of the Earth,
and r is the distance from the center of the Earth to the object.

We also need to know that weight is a measure of the gravitational force acting on an object, so we can equate weight to the force of gravity:

Weight = (G * m * M) / r^2

Now, we need to find out the distance from the surface of the Earth where the weight is 39% of what it was at the surface. Let's call this distance "h."

Weight at the surface (W_surface) = (G * m * M) / r_surface^2

Weight at distance h (W_h) = (G * m * M) / (r_surface + h)^2

We can set up the following equation using the given information:

W_h = 0.39 * W_surface

Substituting the formulas for weight:

(G * m * M) / (r_surface + h)^2 = 0.39 * (G * m * M) / r_surface^2

Canceling out the mass and mass of Earth:

1 / (r_surface + h)^2 = 0.39 / r_surface^2

To simplify the equation, we can cross-multiply:

r_surface^2 = 0.39 * (r_surface + h)^2

Expanding the equation:

r_surface^2 = 0.39 * (r_surface^2 + 2 * r_surface * h + h^2)

Rearranging the equation:

0.61 * r_surface^2 - 0.78 * r_surface * h - 0.39 * h^2 = 0

This is a quadratic equation in terms of h. We can apply the quadratic formula:

h = (-b ± √(b^2 - 4ac)) / (2a)

Where:
a = 0.39
b = -0.78 * r_surface
c = -0.39 * r_surface^2

Plugging in the values, we can calculate h using two significant figures.