Trig Problem?

Question

The problem is:
A spy plane on a practice run over the Midwest takes a picture that shows Cleveland Ohio on the eastern horizon and St. Louis Missouri, 520 miles away on the western horizon. Assuming the radius of the earth is 3950 miles, how high was the plane when the picture was taken.Hint: The sight lines from the plane to the horizons are tangent to the earth and a tangent line to a circle is perpendicular to the radius at the point. The arc of the earth between St. Louis and Cleveland is 520 miles long. Use this fact and the arc length formula to find angle theda

I know the answer is about 8.6 miles but I don't know how to get there

The arc length formula is the radius times the radian measure of the central angle of the arc

Answer 1

If you are required to use the arc length formula then you must realize that this is just an approximation. Then the angle between the two cities is simply 520/3,950, expressed in radians. If the plane is half way between the two cities then the angle theta is half of this. The altitude of the plane will thus be 3,950sec(theta)-3,950, where theta is the 1/2 angle that I descibed. (Remember that the vertex of the angle theta is the center of the Earth.) If you are not allowed to use the secant function then you can calculate it using the pythagorean theorem.

Answer 2

The main points you should notice are
1. Think that plane , St. Louis and Cleveland make a traingele whose only information you know is that the distance of plane from both St. Louis and Cleveland is same.
2. Think that you are at centre of earth then radius of earth is joining both ends of arcmaking the two side of another triange.
3. As the plane and radius of earth are at 90 degree from each other (the fact that sight lines from plane to horizons are tangent)

The above information and arc lenght formula can easily be used to calculate the angle theta. Angle theta should be in degrees and not in miles.