Let's solve the problem step by step.
### Step 1: Understanding the Problem
- We have a plane (P) taking off at a 33-degree angle relative to the observer (O)'s line of sight.
- The plane (P) is directly above a tower (T).
- The distance from the plane (P) to the tower (T) is 7,000 feet.
- We need to find the horizontal distance from the bird (B) circling at a point directly above the observer (O) to the plane (P).
### Step 2: Determining the Relevant Triangle
- We have a right triangle where:
- The angle at the observer's line of sight (O) is [tex]\(33^\circ\)[/tex].
- The distance from the plane (P) to the tower (T) is the adjacent side of the triangle, which is 7,000 feet.
- The distance we need to find, from the bird (B) to the plane (P), is the opposite side of the triangle.
### Step 3: Using Trigonometry
- To determine the distance from the bird to the plane, we use the tangent function, which relates the opposite side to the adjacent side in a right triangle:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, we need to find the opposite side (distance from bird to plane), and we know the angle and the adjacent side (7,000 feet).
### Step 4: Calculation
1. Convert the angle from degrees to radians:
[tex]\[
\text{angle in radians} = 33^\circ \times \left(\frac{\pi}{180}\right) = 0.57596 \text{ radians} \quad (\text{approximately})
\][/tex]
2. Use the tangent function:
[tex]\[
\tan(33^\circ) = \tan(0.57596 \text{ radians})
\][/tex]
So,
[tex]\[
\frac{\text{opposite}}{7000} = \tan(33^\circ)
\][/tex]
From this, we can solve for the opposite side:
[tex]\[
\text{opposite} = 7000 \times \tan(33^\circ)
\][/tex]
3. Using the value from the tangent function:
[tex]\[
\text{opposite} = 7000 \times \tan(33^\circ) \approx 10779.054746702079
\][/tex]
4. Round the result to the nearest whole number:
[tex]\[
\text{distance from bird to plane} \approx 10779 \text{ feet}
\][/tex]
### Conclusion
The distance from the bird (B) to the plane (P) rounded to the nearest whole number is 10,779 feet.
