an airplane over the pacific sights an atoll at a 11 degrees angel angle of depression. if the plane is 410 m above water how many kilometers is it from a point 410 m above the atoll
tan11 = Y/X = 410/X.
X = 410 / tan11 = 2109.3 km.
To find the distance between the plane and the point directly above the atoll, you can use trigonometry and the angle of depression.
Let's label the distance between the plane and the atoll as "x" (in km). Since the angle of depression is 11 degrees, we can consider the tangent of the angle.
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the height above the atoll (410 m) and the adjacent side is the distance between the plane and the point directly above the atoll (x km).
tan(11 degrees) = opposite/adjacent
tan(11 degrees) = 410/x
Now, let's solve for x:
tan(11 degrees) = 410/x
x * tan(11 degrees) = 410
x = 410 / tan(11 degrees)
Using a scientific calculator or an online calculator, we can calculate:
x ≈ 2,450.14 meters
To convert this distance from meters to kilometers, divide by 1000:
x ≈ 2.45 km
Therefore, the distance between the plane and the point directly above the atoll is approximately 2.45 kilometers.
To solve this problem, you can use trigonometric functions and the concept of angles of depression. Here are the steps to find the distance:
1. Draw a diagram representing the situation. Let's label the airplane A, the atoll B, and the unknown point C.
A
|\
h | \
| \
| \
| \
|θ \
| \
| B
|_______|_______
d
2. Assign variables to the given quantities:
- Angle of depression, θ = 11 degrees
- Height of the airplane above the water, h = 410 m
3. From the diagram, we can see that the triangle formed by A, B, and C is a right triangle. The length of BC represents the distance from the point above the atoll.
4. In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Applying this definition, we can write:
tan(θ) = h / d
Rearrange the equation to solve for the distance, d:
d = h / tan(θ)
5. Convert the height from meters to kilometers:
410 m = 410 / 1000 km.
6. Substitute the values into the equation and calculate:
d = (410 / 1000) / tan(11°)
Plug the values into a calculator:
d ≈ 32.967 km
Therefore, the airplane is approximately 32.967 kilometers from the point that is 410 meters above the atoll.