Question

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 while descending at a 12° angle of depression from an altitude of 1,000 feet, we can use some basic trigonometry.

In this situation, we can visualize a right triangle where:
- The altitude of the plane represents the opposite side (which is 1,000 feet).
- The distance the plane still needs to fly represents the hypotenuse.
- The angle of depression from the horizontal to the line of descent (which is also the angle at the ground level to the hypotenuse) is 12°.

We can use the sine function, since the sine of an angle is defined as the ratio of the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In our case:
\[
\sin(12°) = \frac{1000}{d}
\]
where \( d \) is the distance the plane needs to travel (the hypotenuse).

Rearranging the formula to solve for \( d \):
\[
d = \frac{1000}{\sin(12°)}
\]

Now we calculate \( \sin(12°) \):
\[
\sin(12°) \approx 0.2079
\]

Substituting this value back into our equation:
\[
d = \frac{1000}{0.2079} \approx 4811.27
\]

Thus, the distance the plane still needs to fly to reach the ground is approximately \( 4811.27 \) feet.

Rounding this to the nearest hundredths:
\[
\boxed{4811.27} \text{ feet}
\]