An airplane over the Pacific Ocean sights an atoll at an 7° angle of depression. If the plane is 663 m above the water, how many kilometers is it from a point 663 m directly above the atoll?
d/663 = cot 7°
plug and chug
To solve this problem, we can use trigonometry and the concept of angle of depression.
Let's visualize the scenario:
1. Draw a horizontal line to represent the surface of the Pacific Ocean.
2. Draw a vertical line from the atoll to the point directly above it, creating a right triangle.
3. The angle of depression is the angle measured from the horizontal line to the line of sight (line connecting the atoll and the plane).
Now, let's find the distance from the point directly above the atoll to the plane.
We have the following information:
Angle of depression = 7°
Height of the plane above the water = 663 m
The tangent function relates the angle of depression to the opposite side and adjacent side of the right triangle:
tan(angle) = opposite/adjacent
In this case, the opposite side is the height of the plane (663 m) and the adjacent side is the distance from the point directly above the atoll to the plane, which is what we need to find.
Rearranging the formula, we get:
adjacent = opposite / tan(angle)
Plugging in the values:
adjacent = 663 m / tan(7°)
Calculating this, we find the adjacent side to be approximately 5550.20 m.
To convert this distance into kilometers, we divide by 1000:
adjacent = 5550.20 m / 1000 = 5.55020 km
Therefore, the airplane is approximately 5.55020 kilometers away from a point directly above the atoll.