How would you solve this math problem?

Question

Trigonometry can be used to find the circumference of the Earth. From the top of a mountain 5 km high, the angle between the horizon to the top of the mountain is 87.73 degrees. Calculate the radius of the earth.

Answer 1

The idea is that if you're standing on top of a mountain and looking down into the distance at the horizon on flat ground, then your line of vision (the line connecting the mountain point to the part on the earth that's the horizon) is going to be a straight line that's tangent to the earth's surface.

So draw the earth as a circle. Now draw a line from the center of the circle and straight up until it crosses over the edge of the circle. Label the line segment above the circle's edge "5", since this is the height of the mountain. Notice that if r is the earth's radius, then when you're standing on the mountain you are 5+r kilometers away from the very center of the earth. Now from the top of the mountain, draw a diagonal line that passes through this (mountain peak) point and is tangent to the circle. the point where the line touches the circle is the horizon point you see from the mountain.

Draw a line from the earth's center to this horizon point. You've now make a right triangle with a hypotenuse of 5+r and a leg of length r. If the top angle of this triangle is 87.73 degrees, then sin(87.73) = r / (5+r). Now you can solve this for r, and with r you can find the circumference.

Of course, we're assuming a couple things when doing these calculations. We're assuming the earth is perfectly spherical, and that the horizon point is at ground level (where the circumference is measured from).

Answer 2

If you take the vertical through the top of the montain this vertical passes through the center of the earth
The visual from the top to the horizon is tangent to the earth and normal to the radius
sin 87.73 =R/(5+R)
4.996076356=R(0.000784729)
so R= 6366.627 km

Answer 3

E=mc2