Question

A plane is descending at a 12° 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 (2 decimal places).

Answer 1

To find the distance the plane still needs to fly to reach the ground, we can use some basic trigonometry.

We know the following:
- The angle of depression from the horizontal (which is equivalent to the angle of elevation from the ground) is \(12^\circ\).
- The altitude of the plane is \(1,000\) feet.

In this scenario, we can use the tangent function, which relates the angle of a right triangle to the opposite and adjacent sides. The tangent of the angle is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

In our case:
- The "opposite" side of the triangle (altitude) is \(1,000\) feet.
- The "adjacent" side will be the horizontal distance traveled by the plane (\(d\)), which we want to find.

Since we are looking for the distance \(d\) (the ground distance the plane still needs to travel), we use:

\[
\tan(12^\circ) = \frac{1000}{d}
\]

To isolate \(d\), we rearrange the equation:

\[
d = \frac{1000}{\tan(12^\circ)}
\]

Now we need to find \(\tan(12^\circ)\). Using a calculator:

\[
\tan(12^\circ) \approx 0.2126
\]

Substituting this value back into the equation for \(d\):

\[
d = \frac{1000}{0.2126} \approx 4706.29 \text{ feet}
\]

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

\[
\boxed{4706.29} \text{ feet}
\]