Question

A plane is descending at a 12 degree angle of depression. If the current altitude of the plane is 1,000 feet, find the distance the plane still needs to fly to reach the ground. Round to the nearest hundredths.

Answer 1

To find the distance the plane still needs to fly to reach the ground, we can use the concept of right triangles. The altitude of the plane represents the height of the triangle, and the distance the plane needs to fly on the slope forms the hypotenuse of that triangle.

Given:

- Altitude (opposite side in the triangle), \( h = 1000 \) feet
- Angle of depression, \( \theta = 12^\circ \)

Using the sine function, which relates the angle to the opposite side and the hypotenuse, we can set up the following equation:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case, we can rearrange this formula to find the hypotenuse (distance the plane still needs to fly):

\[
\text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)}
\]

Substituting the known values:

\[
\text{hypotenuse} = \frac{1000}{\sin(12^\circ)}
\]

Now, we will calculate \( \sin(12^\circ) \):

\[
\sin(12^\circ) \approx 0.2079
\]

Now substitute this value back into the equation:

\[
\text{hypotenuse} = \frac{1000}{0.2079} \approx 4813.82 \text{ feet}
\]

Thus, the distance the plane still needs to fly to reach the ground is approximately:

\[
\boxed{4813.82 \text{ feet}}
\]