To find the distance \( BP \) (the distance from the bird \( B \) to the plane \( P \)), we can utilize trigonometry and the spatial relationship between the observer \( O \), the tower \( T \), and the plane \( P \).
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Setup the scenario:
- The observer \( O \) is at ground level.
- The tower \( T \) is directly below the plane \( P \).
- The plane \( P \) is at an angle of \( 33^\circ \) from the observer's line of sight to the tower \( T \).
- The distance from the plane \( P \) to the tower \( T \) (horizontal distance) is \( 7000 \) ft.
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Using trigonometry: The angle \( POT \) is \( 33^\circ \), and we can use the cosine function to find the vertical height (height from the tower to the plane) using the formula:
\[ \text{height from } T \text{ to } P (h) = PT \cdot \tan(33^\circ) \]
where \( PT \) is the horizontal distance from the tower to the plane (in this case also the distance that is provided, \( 7000 \) ft).
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Calculate the height \( h \):
\[ h = 7000 \cdot \tan(33^\circ) \]
Now calculating \( \tan(33^\circ) \):
\[ \tan(33^\circ) \approx 0.6494 \text{ (using a calculator)} \]
So,
\[ h = 7000 \cdot 0.6494 \approx 4545.8 \text{ ft} \]
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Finding the total distance \( BP \): Since bird \( B \) is directly above the observer \( O \) and at the same ground level, the total distance from \( B \) to \( P \) can be calculated using the Pythagorean theorem:
\[ BP = \sqrt{PT^2 + h^2} \]
Where \( PT = 7000 \) ft and \( h \approx 4545.8 \) ft.
\[ BP = \sqrt{7000^2 + 4545.8^2} \]
Doing the calculations:
\[ 7000^2 = 49000000 \] \[ 4545.8^2 \approx 20662664 \] \[ BP = \sqrt{49000000 + 20662664} = \sqrt{69662664} \approx 8366.75273 \]
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Rounding to the nearest whole number: Thus,
\[ BP \approx 8367 \text{ ft} \]
However, from the given options, the answer closest to our rounded value is:
8,343 feet.
So, the answer is:
\[ \boxed{8343 \text{ feet}} \]