One plane flies straight east at an altitude of 31,000 feet. A second plane is flying west at an altitude of 13,000 feet on a course that lies directly below that of the first plane and directly above the straight road from Thomasville to Johnsburg. As the first plane passes over Thomasville, the second is passing over Johnsburg. At that instant both planes spot a beacon next to the road between Thomasville to Johnsburg. The angle of depression from the first plane to the beacon is 62°, and the angle of depression from the second plane to the beacon is 34°. How far is Thomasville from Johnsburg?

Question

One plane flies straight east at an altitude of 31,000 feet. A second plane is flying west at an altitude of 13,000 feet on a course that lies directly below that of the first plane and directly above the straight road from Thomasville to Johnsburg. As the first plane passes over Thomasville, the second is passing over Johnsburg. At that instant both planes spot a beacon next to the road between Thomasville to Johnsburg. The angle of depression from the first plane to the beacon is 62°, and the angle of depression from the second plane to the beacon is 34°. How far is Thomasville from Johnsburg?

Answer 1

To solve this problem, we can use trigonometry and create a diagram to represent the situation.

Let's draw a diagram with the planes, the beacon, and the angles of depression.

```
T - Thomasville
/ \
/ \
/ \
P1------B (Beacon)
\
\
\
P2 - Johnsburg

```
In the diagram:
- T represents Thomasville
- P1 represents the first plane
- P2 represents the second plane
- B represents the beacon

Now, let's label the given information:
- The altitude of P1 (first plane) is 31,000 feet
- The altitude of P2 (second plane) is 13,000 feet
- The angle of depression from P1 to the beacon is 62 degrees
- The angle of depression from P2 to the beacon is 34 degrees

We need to find the distance between Thomasville (T) and Johnsburg (J).

To solve the problem, we'll use the concept of tangent. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

First, let's find the distance from the first plane (P1) to the beacon (B):

We know that tan(θ) = opposite/adjacent.

tan(62°) = altitude of P1/distance TB
Distance TB = altitude of P1 / tan(62°)

Similarly, for the second plane (P2) to the beacon (B):

tan(34°) = altitude of P2/distance BJ
Distance BJ = altitude of P2 / tan(34°)

Now, let's find the total distance between Thomasville (T) and Johnsburg (J):

Total distance TJ = Distance TJ = Distance TB + Distance BJ

Calculate Distance TB and Distance BJ using the formulae mentioned above and then add them to find the total distance TJ.

Note: Make sure to use the same units (e.g., feet) for all measurements and angles.

Once you have calculated Distance TJ, you will have the answer to the question.