To find the distance the plane still needs to fly to reach the ground while descending at a 12° angle of depression from an altitude of 1,000 feet, we can use trigonometric functions—specifically, the tangent function.
The angle of depression corresponds to the angle of inclination from the horizontal surface. In this case, we can set up the problem using a right triangle where:
- The altitude (vertical side) is the height of the plane, which is 1,000 feet.
- The distance the plane still needs to fly to reach the ground (the hypotenuse) can be represented as \( d \).
- The angle of depression (which is also the angle of inclination of the hypotenuse with respect to the horizontal) is 12°.
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
We take the opposite side as the altitude (1,000 feet) and the adjacent side as the distance we want to find:
\[ \tan(12°) = \frac{1000}{x} \]
Solving for \( x \):
\[ x = \frac{1000}{\tan(12°)} \]
Now, we can calculate \( \tan(12°) \):
\[ \tan(12°) \approx 0.2126 \quad (\text{using a calculator}) \]
Now, compute \( x \):
\[ x = \frac{1000}{0.2126} \approx 4691.29 \text{ feet} \]
Since this was the horizontal distance, we now need the hypotenuse for the distance the plane needs to fly. To find the hypotenuse \( d \):
Using \( \sin(12°) \):
\[ \sin(12°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1000}{d} \]
So,
\[ d = \frac{1000}{\sin(12°)} \]
Calculating \( \sin(12°) \):
\[ \sin(12°) \approx 0.2079 \]
Now solving for \( d \):
\[ d = \frac{1000}{0.2079} \approx 4811.47 \text{ feet} \]
Rounding this to the nearest foot gives:
\[ \text{Distance the plane still needs to fly} \approx 4811 \text{ feet} \]
Thus, the closest answer is 4810 feet.
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