Question

How tall is a bridge if a 6 foot tall person standing 100 feet away can see the top of the bridge at an angle of 30 degrees to the horizon?

Answer 1

`6 + 100 \ tan 30^circ = 6 + 100/3 sqrt{3}` feet tall.

Explanation:

Let's assume her eyes are at 6 feet. The height `h` and the distance `100` are the legs of a right triangle, `h` opposite `30^circ` so

`tan 30^circ = h/100 or h = 100 tan 30^circ =100/\sqrt{3}= 100/3 sqrt{3}`

The thirty degrees is measured relative to the line where she stared directly across at a spot on the bridge at her eye level. That's the closest point. That way makes a right triangle with legs `100` and `h`. So we have to add six to `h` and say

The bridge is `6 + 100/3 \sqrt{3}` feet tall.