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?
That's quite a long story to state a simple problem
Make a diagram of the side view of the situation.
Let the distance from the "spot beacon" to Thomaville be a, let the distance to the other town be b
the distance between the two towns is a+b
In the Thomaville triangle,
tan 62 = 31000/a
a = 31000/tan62
in the other town triangle
tan34 = 13000/b
b= 13000/tan34
evaluate, add them up, done
Your answer will be in feet, you might need to convert to miles.
( I got 35756.3 ft)
To solve this problem, we can use the concept of trigonometry and the properties of parallel lines.
First, let's visualize the problem. We have two planes flying parallel to each other at different altitudes. At a particular instant, both planes spot a beacon next to the road between Thomasville and Johnsburg. We are given the angle of depression from each plane to the beacon, as well as the altitude of each plane.
Let's break down the problem into smaller parts:
1. Let's label the points involved in the problem. Consider the beacon as point B, Thomasville as point T, and Johnsburg as point J.
2. We know that the angle of depression from the first plane to the beacon is 62°. This means that the line connecting the first plane to the beacon intersects the horizontal line (road) at an angle of 62°.
3. Similarly, the angle of depression from the second plane to the beacon is 34°. This means that the line connecting the second plane to the beacon intersects the horizontal line (road) at an angle of 34°.
4. Let's focus on the triangle formed by the first plane, the second plane, and the beacon. This triangle is a right triangle because the horizontal line (road) acts as the base, and the two lines connecting the planes to the beacon act as the legs.
5. Using trigonometry, we can relate the angle of depression to the altitude. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the altitude (height) of the plane, and the adjacent side is the distance from the plane to the beacon along the horizontal line (road).
6. For the first plane, we have the following equation: tangent(62°) = 31,000 feet / x.
7. For the second plane, we have the following equation: tangent(34°) = 13,000 feet / x.
8. To find the distance from Thomasville to Johnsburg, we need to find the length of the horizontal line (road) between the two planes. Let's call this distance d.
9. Since the planes fly parallel to each other, the lengths of the sides of the triangle formed by the two planes and the beacon are equal. Therefore, the distance from the first plane to the beacon (x) plus the distance from the second plane to the beacon (x) must be equal to the total distance between the two planes (d): x + x = d.
10. Now we have a system of two equations:
- tangent(62°) = 31,000 feet / x
- tangent(34°) = 13,000 feet / x
- x + x = d
11. Solving the first equation for x, we get: x = 31,000 feet / tangent(62°).
Similarly, solving the second equation for x, we get: x = 13,000 feet / tangent(34°).
12. Equating the two expressions for x, we have: 31,000 feet / tangent(62°) = 13,000 feet / tangent(34°).
13. Now we can solve this equation to find the value of x. Once we have x, we can substitute it back into either of the original equations to find the distance d between Thomasville and Johnsburg.
Note: Make sure to convert the angles from degrees to radians if the trigonometric functions in your calculator require input in radians.
By following these steps, you can find the distance between Thomasville and Johnsburg.