To find the distance the plane still needs to fly to reach the ground given a 12-degree angle of depression and a current altitude of 1,000 feet, you can use trigonometry.
The angle of depression is the same as the angle of elevation from the ground to the plane. Thus, we are dealing with a right triangle where:
- The altitude (1,000 feet) is the opposite side to the angle.
- The distance the plane needs to fly is the hypotenuse of the right triangle.
- The angle of elevation (or depression) is 12 degrees.
We can use the sine function, which relates the opposite side to the hypotenuse:
\[ \sin(\theta) = \frac{{\text{opposite}}}{{\text{hypotenuse}}} \]
Here, \(\theta = 12^\circ\), the opposite side is 1,000 feet, and the hypotenuse is the distance we are trying to find (let's call it \(d\)):
\[ \sin(12^\circ) = \frac{1000}{d} \]
Rearranging this gives us:
\[ d = \frac{1000}{\sin(12^\circ)} \]
Now, we can calculate \(\sin(12^\circ)\):
\(\sin(12^\circ) \approx 0.2079\)
Substituting this into the equation:
\[ d = \frac{1000}{0.2079} \approx 4811.5 \]
Rounding to the nearest foot, we find:
\[ d \approx 4812 \text{ feet} \]
Therefore, the distance the plane still needs to fly to reach the ground is approximately 4812 feet.