An airplane approaches a runway at a 3 degree angle of depresssion. If the plane is flying at 30,000 ft., find the ground distance from the airport to the point directly below the plane when the pilot begins the descent. Round your answer to the nearest foot.
To solve this problem, we can use trigonometry. We have an angle of depression of 3 degrees and a height of the airplane at 30,000 ft.
We can use the tangent function to find the ground distance. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the airplane (30,000 ft.) and the adjacent side is the ground distance we are trying to find.
Using the formula tan(angle) = opposite/adjacent, we can rearrange the equation to solve for the adjacent side:
tan(3 degrees) = 30,000 ft. / adjacent
Now, let's solve for the adjacent side:
tan(3 degrees) = 30,000 ft. / adjacent
Using a calculator or the tangent table, we find that the tangent of 3 degrees is approximately 0.0524.
0.0524 = 30,000 ft. / adjacent
To isolate "adjacent," we can multiply both sides of the equation by "adjacent":
0.0524 * adjacent = 30,000 ft.
Dividing both sides of the equation by 0.0524:
adjacent = 30,000 ft. / 0.0524
adjacent ≈ 572519.08 ft.
Rounding to the nearest foot, the ground distance is approximately 572,519 ft.
To find the ground distance from the airport to the point directly below the plane when the pilot begins the descent, we can utilize trigonometry. The angle of depression and the altitude of the plane form a right triangle. We need to find the adjacent side of this triangle, which represents the ground distance.
Given:
- Altitude of the plane, which is 30,000 ft.
- Angle of depression, which is 3 degrees.
To calculate the ground distance, we can use the tangent function:
tan(angle) = opposite / adjacent
In this case, the opposite side is the altitude of the plane, and the adjacent side is the ground distance that we are trying to find.
tan(3 degrees) = 30,000 ft / ground distance
We can rearrange the equation to solve for ground distance:
ground distance = 30,000 ft / tan(3 degrees)
Now, let's calculate the ground distance using a calculator or a programming language that supports trigonometric functions:
tan(3 degrees) ≈ 0.0524
ground distance ≈ 30,000 ft / 0.0524 ≈ 571,224 ft.
Therefore, the ground distance from the airport to the point directly below the plane when the pilot begins the descent is approximately 571,224 feet.