To find the distance the plane still needs to fly to reach the ground while descending at an angle of depression of 12 degrees from an altitude of 1,000 feet, we can use trigonometry.
In this scenario, we can represent the situation with a right triangle where:
- The altitude (vertical leg) is 1,000 feet.
- The angle of depression is 12 degrees, which corresponds to the angle of elevation from the point on the ground directly below the plane.
We need to find the hypotenuse of this triangle, which represents the distance the plane needs to fly to reach the ground.
Using the sine function, we have:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In our case:
- \(\theta = 12^\circ\)
- The opposite side (altitude) = 1,000 feet
- Hypotenuse = \(d\) (the distance to the ground)
We can rearrange the sine equation to find \(d\):
\[ d = \frac{\text{opposite}}{\sin(\theta)} = \frac{1000}{\sin(12^\circ)} \]
Now, we can calculate \(\sin(12^\circ)\):
\[ \sin(12^\circ) \approx 0.2079 \]
Now substituting back into the equation for \(d\):
\[ d = \frac{1000}{0.2079} \approx 4815.12 \]
Now rounding to the nearest foot:
\[ d \approx 4815 \text{ feet} \]
Therefore, the distance the plane still needs to fly to reach the ground is approximately 4815 feet.
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