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A pilot approaching a 3000-meter runway finds that the angles of depression to the ends of the runway are 14° and 20°. How far is the plane from the closer end of the runway?

Round to the nearest tenth place.

Answer :

Final answer:

To find out how far the plane is from the closer end of the runway, we use the angles of depression and runway length to calculate the distance using trigonometric ratios. The distance is found to be approximately 26153 meters.

Explanation:

To determine the distance from the plane to the closer end of the runway, we use trigonometric ratios based on the angles of depression given. The angles of depression to the ends of the runway are 14° and 20°. Since the problem involves a horizontal surface (runway) and vertical angles (angles of depression), we can create two right triangles. One triangle has an angle of 14° between the plane's line of sight and the horizontal surface, and the other one has a 20° angle.

Let x be the distance from the plane to the point directly above the closer end of the runway and y be the distance from the plane to the point directly above the far end of the runway. We are looking for x, which can be found using the tangent of the 14° angle.

x = runway length / (tan(20°) - tan(14°))

Given the runway length as 3000 meters, we can calculate:

x = 3000 / (tan(20°) - tan(14°))

Crunching the numbers:

x ≈ 3000 / (0.3640 - 0.2493)

x ≈ 3000 / 0.1147

x ≈ 26153 meters (rounded to the nearest tenth)

Therefore, the distance from the plane to the closer end of the runway is approximately 26153 meters.

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