To determine the length of the rescue line, we can use trigonometry.
Let's denote:
- \( h = 40 \) m (height of the helicopter above the surface of the sea),
- \( \theta = 53.13^\circ \) (angle of depression from the helicopter to the swimmer),
- \( L \) (length of the rescue line).
When we form a right triangle from the helicopter to the swimmer, we have:
- The height of the helicopter forms one side of the triangle (opposite to the angle of depression).
- The length of the rescue line forms the hypotenuse of the triangle.
Since the angle of depression equals the angle of elevation from the swimmer's perspective, we can use the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L} \]
Rearranging this formula gives us:
\[ L = \frac{h}{\sin(\theta)} \]
Now we can substitute \( h = 40 \) m and \( \theta = 53.13^\circ \):
First, we calculate \( \sin(53.13^\circ) \):
\[ \sin(53.13^\circ) \approx 0.8 \]
Now we can plug in the values:
\[ L = \frac{40}{0.8} = 50 \text{ m} \]
Therefore, the length of the rescue line is approximately 50 meters.
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