SPRECALC7 6.2.054.

Question

A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 30,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is 23°. (Round your answers to the nearest foot.)

Answer 1

It's helpful to draw a sketch of the situation.

Draw the plane. Draw the ground, a horizontal line below the plane. Then draw a line straight down from the plane to ground and mark that as 30,000 feet.

Then to draw a horizontal line (up in the sky) through the plane. And draw a diagonal line from the plane to ground, down and to the right, 23^o below the top line. (The angle doesn't have to be exact.) That is angle of depression. Where the diagonal line meets the ground is where the arch is.

Now, you have triangle between the plane and two points on the ground. The angle at the top of the triangle is 90-23 = 67^o

The distance from the plane to the arch is along that diagonal line, the hypothenuse of the triangle. Using the SOH CAH TOA, we have cos (67) = 30,000 / H. Calculate H and that's the answer. I get 76,779 feet.