an airplane is flying at a height of 2 miles above the level ground. the angle of depression from the plane to the foot of a tree is 15 degree. Find the distance that the plane must fly to be directly above the tree.
tan 15° = 2/x
x = 2/tan15 = 7.464 miles
To solve this problem, we can use trigonometry and apply the tangent function.
Let's label the following:
- The height of the airplane above the ground as h = 2 miles
- The angle of depression as θ = 15 degrees
We need to find the horizontal distance, which we'll call x, that the plane must fly to be directly above the tree.
By using the tangent function, we have the following relationship:
tan(θ) = opposite / adjacent
In this case, the opposite side is the height of the airplane (h = 2 miles) and the adjacent side is the horizontal distance (x) that the plane must fly.
tan(15 degrees) = 2 / x
To find the value of x, we can rearrange the equation:
x = 2 / tan(15 degrees)
Now, we can calculate the value of x using a calculator:
x ≈ 7.41 miles
Therefore, the distance that the plane must fly to be directly above the tree is approximately 7.41 miles.
To solve this problem, we can use trigonometric ratios.
Let's draw a diagram to better visualize the problem:
```
|
A |
|\
| \
T | \
| \
| \
-------
```
In the diagram:
- A represents the airplane
- T represents the foot of the tree
- The line connecting the airplane to the tree is labeled as AT
- The angle of depression from the plane to the foot of the tree is labeled as ∠ATP (where P is a point directly below A)
We are given:
- The height of the airplane (AP) is 2 miles above the level ground.
- The angle of depression (∠ATP) is 15 degrees.
We need to find the distance that the plane must fly to be directly above the tree, which is the length of TP.
To find TP, we can use the tangent of angle ∠ATP.
In trigonometry, tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle.
So, we have:
tan(∠ATP) = opposite side / adjacent side
In our case, the opposite side is AP (2 miles) and the adjacent side is TP (the distance we want to find).
Therefore, we can write:
tan(∠ATP) = AP / TP
Substituting the values into the equation:
tan(15°) = 2 / TP
To find TP, we need to isolate it on one side of the equation.
Rearranging the equation, we have:
TP = 2 / tan(15°)
Calculating this on a scientific calculator or online calculator, we find:
TP ≈ 8.583 miles
Therefore, the distance that the plane must fly to be directly above the tree is approximately 8.583 miles.