Final answer:
To determine the height of the hot air balloon, the tangent of the angle of elevation (28 degrees) is set equal to the ratio of the height of the balloon to the distance from the observer (50 meters), and then the height is found using basic trigonometric operations.
Step-by-step explanation:
To find the height of the hot air balloon given the distance from the observer to the balloon and the angle of elevation, we use trigonometry. Specifically, we use the tangent function which relates the angles of a right triangle to the ratios of two of its sides. Here, the angle given is the angle of elevation (28 degrees) and the adjacent side is the distance from the observer to the balloon (50 meters).
The relationship can be written as the tangent of angle = opposite side / adjacent side. So we have, tan(28 degrees) = height / 50 meters. To solve for the height, we multiply both sides by 50 meters, which gives us the equation: height = 50 meters * tan(28 degrees).
By performing this calculation, we determine the height of the hot air balloon. As this is a basic trigonometry problem, it demonstrates the practical application of math to solve real-world geometry problems.