The angle of depression from a descending airplane to the control tower below is 60∘.
If the plane is 35,000 feet above the ground, approximately what is the plane's horizontal distance from the control tower?
70,000 ft
20,207 ft
24,749 ft
60,622 ft
To find the plane's horizontal distance from the control tower, we can use trigonometry. We will use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle.
Let's label the angle of depression as A, the distance from the plane to the control tower as x, and the height of the plane as h.
From the information given, the height of the plane (opposite side) is 35,000 ft and the angle of depression is 60∘.
Using the tangent function, we have:
tan(A) = opposite/adjacent
tan(60∘) = 35,000/x
To solve for x, we can rearrange the equation:
x = 35,000 / tan(60∘)
Using a calculator, the tangent of 60∘ is √3. Therefore:
x = 35,000 / √3
Calculating this, we find:
x ≈ 20,207 ft
Therefore, the plane's horizontal distance from the control tower is approximately 20,207 ft.
So the correct answer is 20,207 ft.
To find the horizontal distance from the control tower, we can use trigonometry.
The angle of depression is the angle measured from the horizontal line to the line of sight down to the control tower. In this case, the angle of depression is given as 60°.
We can use the tangent function to calculate the horizontal distance. Tangent is defined as the ratio of the opposite side to the adjacent side of a right triangle.
In this case, the opposite side is the height of the plane above the ground (35,000 ft) and the adjacent side is the horizontal distance we want to find.
So, we have:
tan(60°) = (opposite side) / (adjacent side)
tan(60°) = 35,000 ft / (adjacent side)
To find the adjacent side (horizontal distance), we can rearrange the equation:
(adjacent side) = (opposite side) / tan(60°)
(adjacent side) = 35,000 ft / (tan(60°))
Using a calculator, we can calculate the tangent of 60°:
tan(60°) ≈ 1.732
Now, plug in the values:
(adjacent side) = 35,000 ft / 1.732
(adjacent side) ≈ 20,207 ft
Therefore, the approximate horizontal distance from the control tower is 20,207 ft.
So, the correct answer is 20,207 ft.
tan 60 = 35,000 / h
h = 35,000 / tan 60