Math Question that I need help on

Question

You are standing about 6 km above sea level at the edge of the peak of Tanzania's Mount Kilimanjaro, the highest freestanding mountain on Earth. From the peak (point P), you look down to your right to see point X on the horizon. Down to your left, you see up to point Y on the horizon. Beyond X and Y, you can not see anything because your view is blocked by the Earth, so XY⌢ represents the portion of the Earth which you can see. Using the fact that the radius of the Earth is about 6400 km, find the measure of XY⌢ to the nearest integer

Answer 1

Start by drawing a picture.

From the top of the 6km high mountain, you can draw 2 lines that are tangents to the circle of the Earth. Each tangent is one leg of a right triangle, with the perpendicular leg being the radius of the Earth.

The hypotenuse of this right triangle is the radius of the Earth plus the the height of the mountain.

Knowing one leg and the hypotenuse, us the Pythagorean Theorem to get the distance from the top of the mountain to the tangent point:

d² = 6400² + 6406²