To find the distance the plane still needs to fly to reach the ground while descending at a 12° angle and starting from an altitude of 1,000 feet, we can use some basic trigonometry.
The angle of depression and the height can be related using the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case:
- The "opposite" side is the altitude of the plane, which is 1,000 feet.
- The "adjacent" side is the horizontal distance the plane will travel before it reaches the ground.
Since we want to find the hypotenuse (the distance the plane needs to fly), we can use the sine function, which relates the angle to the opposite side and hypotenuse:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Rearranging the formula gives us:
\[ \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} \]
Plugging in the values:
\[ \text{hypotenuse} = \frac{1,000 \text{ feet}}{\sin(12^\circ)} \]
Now, let's calculate \( \sin(12^\circ) \):
Using a calculator:
\[ \sin(12^\circ) \approx 0.2079 \]
Now substituting this back into the hypotenuse formula:
\[ \text{hypotenuse} = \frac{1,000}{0.2079} \approx 4811.34 \text{ feet} \]
Thus, the distance the plane still needs to fly to reach the ground is approximately:
\[ \boxed{4811.34} \text{ feet} \]
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