The problem you have uploaded describes two airplanes flying vertically above a point on the ground at an altitude of 3000 meters, with observation angles of 60 degrees and 45 degrees from the same point on the ground. The task is to find the horizontal distance between the two airplanes.

We can break this down into a trigonometry problem, where the goal is to determine the difference in the horizontal distances of the two airplanes relative to the observer on the ground. Here's the method to solve it:

Step-by-Step Solution:

1. Identify Known Values:

   • Height (altitude) of the airplanes = 3000 m.
   • Angle of elevation to one airplane = 60°.
   • Angle of elevation to the second airplane = 45°.

2. Trigonometry Setup:

   • From the diagram, the situation forms two right-angled triangles. The height of both triangles is the altitude of the airplanes, 3000 m.
   • We need to find the horizontal distances from the observation point to each airplane using the formula for a right triangle: $\text{Horizontal Distance} = \frac{\text{Height}}{\tan(\theta)}$ where $\theta$ is the angle of elevation.

3. Calculate Horizontal Distance for Each Airplane:

   • For the airplane at a 60° angle: $d_1 = \frac{3000}{\tan(60^\circ)} = \frac{3000}{\sqrt{3}} = 1732.05 \, \text{meters}$
   • For the airplane at a 45° angle: $d_2 = \frac{3000}{\tan(45^\circ)} = \frac{3000}{1} = 3000 \, \text{meters}$

4. Calculate the Distance Between the Two Airplanes:

   • The horizontal distance between the two airplanes is the difference between $d_2$ and $d_1$: $\text{Distance} = d_2 - d_1 = 3000 - 1732.05 = 1267.95 \, \text{meters}$

Final Answer:

The horizontal distance between the two airplanes is approximately 1268 meters.

Would you like further clarification or more details?

Here are 5 related questions to deepen understanding:

1. What other types of problems use angles of elevation and distances?
2. How would the calculation change if the airplanes were at different altitudes?
3. What is the difference between angles of elevation and angles of depression in such problems?
4. Could we use sine or cosine rules to solve this problem? Why or why not?
5. What happens to the distance between the airplanes if both angles of elevation were the same?

Tip: Always check if you are dealing with right-angled triangles when solving for distances using trigonometry. It simplifies the problem greatly!