Question

Kayla spots an airplane on radar that is currently approaching in a straight line, andthat will fly directly overhead. The plane maintains a constant altitude of 6875 feet.Kayla initially measures an angle of elevation of 16° to the plane at point A. At somelater time, she measures an angle of elevation of 30° to the plane at point B. Find thedistance the plane traveled from point A to point B. Round your answer to thenearest foot if necessary.

Answer 1

Given:

The altitude of the plane

$=6875\text{ feet.}$

The two angles of elevations when the plane is at point A and B are

$16\degree\text{ and }30\degree.$

Required:

We have to find the distance traveled by the plane from point A to point B.

Explanation:

We will solve this question in two cases.

Case 1:

When the point A and B are on the same side of Kyla:

In this case, we have to find the distance of the foot to the altitude of the plane from Kayla when the plane is at points A and B and then subtract.

The distance of the foot to the altitude of the plane from Kayla when the plane is at point A

$D_1=\frac{6875}{tan16\degree}=\frac{6875}{0.287}=23975.97\text{ feet.}$

The distance of the foot to the altitude of the plane from Kayla when the plane is at point B

$D_2=\frac{6875}{tan30\degree}=\frac{6875}{\frac{1}{\sqrt{3}}}=6875\sqrt{3}=11907.85\text{ feet.}$

Therefore, the distance traveled by the plane from point A to point B

$\begin{gathered} =23975.97-11907.85 \\ =12068.12\text{ feet.} \end{gathered}$

Case 2:

When the point A and B are on the different sides of Kyla:

In this case, we have to find the distance of the foot to the altitude of the plane from Kayla when the plane is at points A and B and then add.

Therefore, the distance traveled by the plane from point A to point B

$\begin{gathered} =23975.97+11907.85 \\ =35883.83\text{ feet.} \end{gathered}$

Final answer:

Hence the final answer is:

When the point A and B are on the same side of Kyla

$=12068\text{ feet. \lparen Round to the nearest foot\rparen}$

When the point A and B are on the different sides of Kyla

$=35884\text{ feet. \lparen Round to the nearest foot\rparen}$