8.2.1 Angles of Elevation and Depression (continued)
Example 3 Finding Distance by Using Angle of Depression
A horizontal distance is calculated here using an angle of depression. The angle of depression is given. Also, the altitude of the viewing point, or the length of the leg opposite the angle of depression, is given. The angle of depression is 9 ° . The height from the fire to the horizontal line is 70 feet. Because the horizontal and the ground-level lines are assumed to be parallel, the angle formed by the line of sight and the ground is also 9 ° , by the Alternate Interior Angles Theorem. If x is the horizontal distance from the tower to the fire, then tan 9 ° = 70/ x (the opposite leg over the adjacent leg). Solving for x yields a distance of approximately 442 feet. A horizontal distance is calculated here using two angles of depression. The angles of depression are given. Also, the altitude of the viewing point is given. The strategy is to find the distance from the observer to each of the control towers, and then to find the distance between the towers by subtraction. The measures of the angles opposite the ground leg of the triangles are 42 ° and 51 ° by the Alternate Interior Angles Theorem. This step is not necessary since, because the horizontal and the ground level are assumed to be parallel, the upper triangles may be used just as well. If a is the horizontal distance from the jet to one tower, then tan 51 ° = 1.9/ a . Solving for a yields a distance of approximately 1.538 km. If b is the horizontal distance from the jet to the other tower, then tan 42 ° = 1.9/ b . Solving for b gives an
Example 4 Aviation Application
approximate distance of 2.110 km. The distance between the towers is b − a = 2.110 − 1.538 ≈ 0.6 km.
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