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

To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles. The distance the plane traveled from point A to point B is approximately 24114.91 feet.

Step-by-step explanation:

To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles. Let's assume that point A is directly below the plane and point B is on the ground, forming a right triangle. The angle of elevation at point A is 16° and the angle of elevation at point B is 30°. The distance between point A and point B is the base of the triangle.

We can set up a proportion using the tangent function:

tan(16°) = 6875 / x

Solving for x, we can rearrange the equation as:

x = 6875 / tan(16°)

Using a calculator, we find that x ≈ 24114.91 feet. Therefore, the distance the plane traveled from point A to point B is approximately 24114.91 feet.

Answer 2

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.

Step-by-step 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=(6875)/(tan16\degree)=(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=(6875)/(tan30\degree)=(6875)/((1)/(√(3)))=6875√(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}`