To find the horizontal distance from a point directly beneath the plane to the airport tower, we can use the concept of right triangles and the tangent function from trigonometry.
Given:
- Altitude of the plane (height above ground): \( h = 12,000 \) m
- Angle of depression from the pilot to the airport tower: \( \theta = 32^\circ \)
In the context of the problem, the angle of depression is equal to the angle of elevation from the airport tower to the plane. Therefore, we can use the tangent function, which relates the angle to the opposite side (height of the plane) and the adjacent side (horizontal distance we want to find).
The formula is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
where:
- \( h = 12,000 \) m (height of the plane)
- \( d \) is the distance from the point directly beneath the plane to the airport tower.
Rearranging the formula to solve for \( d \):
\[ d = \frac{h}{\tan(\theta)} \]
Substituting in the values we have:
\[ d = \frac{12,000 , \text{m}}{\tan(32^\circ)} \]
Now, we can calculate \( \tan(32^\circ) \):
Using a calculator:
\[ \tan(32^\circ) \approx 0.6249 \]
Substituting this value back into the equation:
\[ d = \frac{12,000 , \text{m}}{0.6249} \approx 19,205 , \text{m} \]
Rounding to the nearest whole number, the distance from a point directly beneath the plane to the airport tower is approximately:
\[ \boxed{19205} , \text{m} \]