Two planes are flying at the height of 10-km. They are going at the same speed, and both traveling due north. If the planes are 3,000 km apart as measured along the surface of the earth, what angle do the lines from the planes make to the center of the Earth make?

You can see that the altitude of the planes makes no difference. If the earth is considered a sphere with a polar radius of 6357 km, then an arc of 3000 km subtends an angle of

3000/(2pi*6357) radians

To find the angle that the lines from the planes make to the center of the Earth, we can use some trigonometry.

First, let's draw a diagram to visualize the situation. Draw a circle to represent the Earth, and draw two lines from the planes to the center of the Earth. Since the planes are flying due north, the lines will be vertical.

The planes are 3,000 km apart as measured along the surface of the Earth. This means that the distance between the two lines is a chord of the circle, where the Earth's radius is the distance from the center of the Earth to the surface.

Using the Pythagorean theorem, we can calculate the radius of the Earth. Let's call it "r."

r^2 + 10^2 = (r + 10)^2

Expanding this equation, we have:

r^2 + 100 = r^2 + 20r + 100

Simplifying, we get:

20r = 100

r = 5 km

Now that we know the radius of the Earth, we can find the angle that the lines from the planes make to the center of the Earth. This angle is the same as the angle subtended by the chord (the distance between the two lines) from the center of the Earth.

Using trigonometry, we can find this angle by taking the inverse sine of the ratio of the chord length to the diameter of the circle.

In this case, the chord length is 3,000 km, and the diameter is twice the radius of the Earth (2 * 5 km = 10 km).

So, the angle is:

angle = arcsin(chord length / diameter)
angle = arcsin(3,000 km / 10 km)

Calculating this using a calculator, we find that the angle is approximately 89.85 degrees.

Therefore, the angle that the lines from the planes make to the center of the Earth is approximately 89.85 degrees.