A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon? 


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Verified step by step guidance

1. Identify the key elements in the problem: the balloonist's height above the ground (h), the distance between the two villages (1.5 miles), and the angles of depression to the closer village (35°) and the farther village (31°).
2. Understand that the angles of depression correspond to the angles of elevation from the ground to the balloon when viewed from the villages. Thus, the angles of elevation are also 35° and 31° respectively.
3. Visualize or draw a diagram with the balloon directly above the midpoint of the line joining the two villages. Label the closer village as point A, the farther village as point B, and the balloon's position vertically above the midpoint as point C. The line segment AB represents the 1.5 miles.
4. Use trigonometric ratios to set up equations. For the triangle formed by the balloon, the midpoint, and each village: use the tangent function, which relates the opposite side (height of the balloon, h) to the adjacent side (half the distance from the midpoint to each village, 0.75 miles). Set up the equations tan(35°) = h / 0.75 and tan(31°) = h / 0.75.
5. Solve the system of equations to find the height 'h' of the balloon. Since both equations should ideally give the same height, you can average the results from both equations to find the most accurate height of the balloon.

Key Concepts

Angle of Depression

The angle of depression is the angle formed by a horizontal line from an observer's eye to an object below the horizontal line. In this scenario, the balloonist observes two towns at different angles of depression, which helps determine the height of the balloon. Understanding this concept is crucial for visualizing the problem and applying trigonometric functions.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this problem, the tangent function is particularly useful, as it relates the height of the balloon to the distances to the towns based on the angles of depression. Mastery of these ratios is essential for solving problems involving right triangles.

Right Triangle Properties

The properties of right triangles are foundational in trigonometry, as they allow for the application of trigonometric ratios. In this case, the balloon's height, the distances to the towns, and the angles of depression form two right triangles. Recognizing and applying these properties enables the calculation of unknown lengths and heights in the context of the problem.