To find the distance the plane still needs to fly to reach the ground, we can use trigonometry. The problem gives us the altitude and the angle of depression.
The relationship between the altitude (height), the angle of depression (angle with respect to the horizontal), and the distance to be traveled (hypotenuse) can be represented in a right triangle.
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Identify the components of the triangle:
- Altitude (opposite side) = 1000 feet
- Angle of depression = 12°
- Distance the plane needs to fly = hypotenuse (let's denote it as \(d\))
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Using the sine function: The sine function relates the angle to the opposite side and hypotenuse: \[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \] Here, we use the angle of depression, which is 12°, and the altitude (1000 feet): \[ \sin(12°) = \frac{1000}{d} \]
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Rearranging to solve for \(d\): \[ d = \frac{1000}{\sin(12°)} \]
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Calculating \(\sin(12°)\): Using a calculator: \[ \sin(12°) \approx 0.2079 \]
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Plugging it back into the equation: \[ d = \frac{1000}{0.2079} \approx 4811.55 \text{ feet} \]
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Rounding to the nearest foot: \[ d \approx 4812 \text{ feet} \]
Therefore, the distance the plane still needs to fly to reach the ground is approximately 4812 feet.