Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Height of a Tower The angle of depression from a television tower to a point on the ground 36.0 m from the bottom of the tower is 29.5°. Find the height of the tower.

Verified step by step guidance

1. Draw a right triangle representing the situation: the tower is the vertical side (height h), the distance from the tower to the point on the ground is the horizontal side (36.0 m), and the line of sight from the top of the tower to the point on the ground forms the hypotenuse.
2. Identify the angle of depression (29.5°) as the angle between the horizontal line from the top of the tower and the line of sight down to the point on the ground. This angle is congruent to the angle between the ground and the line of sight inside the triangle.
3. Use the tangent function, which relates the opposite side (height of the tower h) to the adjacent side (distance 36.0 m) in a right triangle: \(\tan(29.5^\circ) = \frac{h}{36.0}\).
4. Rearrange the equation to solve for the height h: \(h = 36.0 \times \tan(29.5^\circ)\).
5. Calculate the value of \(\tan(29.5^\circ)\) using a calculator and multiply by 36.0 to find the height of the tower.

Key Concepts

Angle of Depression

The angle of depression is the angle formed between the horizontal line from the observer's eye and the line of sight to an object below the horizontal. In this problem, it helps relate the observer's viewpoint at the top of the tower to the point on the ground, allowing the use of trigonometric ratios.

Right Triangle Trigonometry

The problem involves a right triangle formed by the tower height, the horizontal distance from the tower base, and the line of sight. Using trigonometric functions like tangent, which relates opposite and adjacent sides, allows calculation of the unknown height.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Here, tan(29.5°) = height / 36.0 m, which can be rearranged to find the tower's height by multiplying the tangent of the angle by the horizontal distance.