A pilot of an airplane flying at 12,000 feet sights a water tower. The angle of depression to the base of the tower is 25°. What is the length of the line of sight from the plane to the tower?
12,000/sin 25
To find the length of the line of sight from the plane to the tower, we can use trigonometry and the concept of angles of elevation and depression.
First, let's draw a diagram to visualize the situation:
|
| 12,000 ft
|
| - - - - line of sight
|
|--------- water tower (base)
In this scenario, the angle of depression is the angle formed between the horizontal line and the line of sight from the plane to the base of the tower. We are given that the angle of depression is 25°.
Now, let's use trigonometry to find the length of the line of sight. We can use the tangent function, which relates the angle of depression to the opposite and adjacent sides of a right triangle.
In this case, the opposite side is the height of the plane (12,000 ft), and the adjacent side is the length of the line of sight.
Tangent(theta) = Opposite / Adjacent
Tangent(25°) = 12,000 ft / Adjacent
To find the length of the line of sight (Adjacent), we rearrange the equation:
Adjacent = 12,000 ft / Tangent(25°)
Using a scientific calculator, we can find the tangent of 25°, which is approximately 0.4663.
Adjacent = 12,000 ft / 0.4663
Adjacent ≈ 25,742.48 ft
Therefore, the length of the line of sight from the plane to the tower is approximately 25,742.48 feet.
To solve this problem, we can use trigonometry. Since we have the angle of depression and the height of the airplane, we can find the length of the line of sight.
Step 1: Draw a diagram to represent the situation. Let's label the height of the airplane as "h" (12,000 feet) and the distance from the airplane to the water tower as "x". The angle of depression is 25°.
A
|¯¯¯¯¯¯¯|
| |
| |
| |
x |___25°|
T
|\
| \ h
| \
| \
Step 2: We can use the tangent function to find the ratio of the opposite (h) to the adjacent (x) side of the right triangle formed.
tan(25°) = h/x
Step 3: Rearrange the formula to solve for x.
x = h / tan(25°)
Step 4: Substitute the values into the equation.
x = (12,000 ft) / tan(25°)
Using a calculator, tan(25°) = 0.4663 (rounded to four decimal places).
x = (12,000 ft) / 0.4663
x ≈ 25,757.89 ft
The length of the line of sight from the plane to the water tower is approximately 25,757.89 feet.