Final answer:
The height of the balloon can be calculated using trigonometry by setting up a system of equations based on the angles of depression to consecutive mileposts and solving for the height, h.
Step-by-step explanation:
To calculate the height of the balloon above the ground using the angles of depression to two consecutive mileposts, we can use trigonometry. Let's assume the distance between the mileposts is one mile (5280 feet), and we'll denote the smaller angle of depression as α (alpha) = 22° and the larger one as β (beta) = 26°.
Consider the triangle formed by drawing lines from the balloon to each of the mileposts and the line joining the mileposts. Since angles of depression are equal to the angles of elevation when measured from the ground, we have two right triangles sharing a common side (the height of the balloon).
Let h be the height of the balloon and d be the horizontal distance from the balloon to the closer milepost. Using the tangent function:
Tan(α) = h / d
Tan(β) = h / (d + 5280)
We now have a system of two equations with two variables. Solving for h, we find the height of the balloon in feet is at the point where both equations yield the same value for h.