To find the distance the plane needs to fly to reach the ground while descending at a 12-degree angle of depression from an altitude of 1,000 feet, we can use trigonometry.
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Identify the angles and sides:
- Let \( h = 1000 \) feet be the altitude (vertical side).
- Let \( d \) be the horizontal distance to the point directly below the plane when it reaches the ground.
- The angle of depression is \( 12^\circ \), which is also the angle of elevation from the ground to the plane.
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Use the tangent function: The tangent of the angle in a right triangle is the ratio of the opposite side (altitude) to the adjacent side (horizontal distance): \[ \tan(12^\circ) = \frac{h}{d} \]
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Rearranging the formula gives us: \[ d = \frac{h}{\tan(12^\circ)} \]
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Calculate \( d \): First, we need to calculate \( \tan(12^\circ) \): \[ \tan(12^\circ) \approx 0.2126 \quad (\text{using a calculator}) \]
Now, substitute the value of \( h \): \[ d = \frac{1000}{\tan(12^\circ)} = \frac{1000}{0.2126} \approx 4691.14 \text{ feet} \]
Thus, the distance the plane still needs to fly to reach the ground, rounded to the nearest hundredths, is:
\[ \boxed{4691.14} \text{ feet} \]