Charlie is late for the hot-air balloon rally. He stops his car 65 meters from a hot-air balloon already in flight. While Charlie watches the balloon ascend, he tilts his eyes up from 28º until he is watching at an angle of elevation of 66º. Find the vertical distance to the nearest tenth of a meter that the balloon ascended while Charlie watched.
To solve this problem, we can use trigonometry and the concept of similar triangles.
Let's break down the information given:
- Charlie stops his car 65 meters from the hot-air balloon, forming a right angle with the ground.
- Charlie's line of sight to the balloon starts at an angle of elevation of 28º.
- Charlie continues to watch the balloon ascend until his line of sight reaches an angle of elevation of 66º.
We are asked to find the vertical distance that the balloon ascended while Charlie was watching.
First, let's draw a diagram to visualize the situation:
```
A
/ |
/ | <- Distance = 65 meters
/ |
Charlie (C) / |
/ | x (vertical distance)
/ |
/ |
/ |
/_________|
B
```
In the diagram, point A represents the hot-air balloon, point B is where Charlie stopped his car, and point C is Charlie's position. The vertical distance we need to find is represented by the variable 'x'.
From the diagram, we can see that right triangle ABC is formed, where the angle at point C is 90º.
Now, let's use trigonometry to solve for 'x':
In triangle ABC, we have:
- Angle ACB = 28º (initial angle of elevation)
- Angle BAC = 66º (final angle of elevation)
- Distance AB = 65 meters
We can use the tangent function to find the vertical distance 'x':
tan(angle) = opposite/adjacent
In triangle ABC, the vertical distance 'x' is the opposite side, and the distance AB is the adjacent side.
Using the tangent function:
tan(28º) = x/65
To find 'x', we can rearrange the equation as follows:
x = 65 * tan(28º)
Using a scientific calculator, we can calculate:
x ≈ 65 * tan(28º) ≈ 38.78 meters.
Therefore, the vertical distance that the balloon ascended while Charlie watched is approximately 38.78 meters to the nearest tenth of a meter.