To find the horizontal distance from the point directly beneath the plane to the airport tower, we can use trigonometry, specifically the definition of the tangent function.
The problem involves a right triangle where:
- The altitude of the plane (the vertical side) is 12,000 m.
- The angle of depression from the plane to the airport tower is 32 degrees.
The angle of depression is equal to the angle of elevation from the airport tower to the plane. So we can use the tangent function, which relates the opposite side (the altitude) to the adjacent side (the horizontal distance).
Let's denote:
- \( h = 12,000 \) m (the height of the plane)
- \( d \) = the horizontal distance from the tower to the point directly underneath the plane
- \( \theta = 32 \) degrees (the angle of depression)
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
Rearranging for \( d \):
\[ d = \frac{h}{\tan(\theta)} \]
Now substituting the known values:
\[ d = \frac{12,000}{\tan(32^\circ)} \]
Calculating \( \tan(32^\circ) \) (using a calculator):
\[ \tan(32^\circ) \approx 0.6249 \]
Now substituting that value back into the equation:
\[ d = \frac{12,000}{0.6249} \approx 19,189.7 \text{ m} \]
So, the horizontal distance from the tower to the point directly beneath the plane is approximately:
\[ \boxed{19,190 \text{ m}} \]
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