A man is flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression from his balloon to a friend's car in the parking lot is 38°. A minute and a half later, after flying directly over this friend's car, he looks back to see his friend getting into the car and observes the angle of depression to be 30°. At that time, what is the distance between him and his friend?
Let's assume that the man is at point B, directly above his friend's car at point C, and his friend is at point A.
When the angle of depression is 38°, triangle ABC is formed. We can use trigonometry to find the height of the balloon, BC.
tan(38°) = BC / AB
BC = AB * tan(38°)
BC = AB * 0.7813
Similarly, when the angle of depression is 30°, triangle ABD is formed. We can use trigonometry to find the height of the balloon, BD.
tan(30°) = BD / AB
BD = AB * tan(30°)
BD = AB * 0.5774
Let's calculate the time it takes for the balloon to fly from point B to point A.
Given that the balloon flies at a constant rate of 5 feet per second and it takes a minute and a half to fly from B to A, the time taken is 1.5 minutes * 60 seconds/minute = 90 seconds.
During this time, the balloon flies a distance equal to its rate of travel multiplied by the time.
Distance = Rate * Time
Distance = 5 ft/s * 90 s
Distance = 450 ft
Now, we can calculate the distance between the man and his friend at point A.
Using the Pythagorean theorem, we have the following equation:
AB^2 = BC^2 + AC^2
Substituting the known values, we get: AB^2 = (AB * 0.7813)^2 + 450^2
Simplifying, we have: AB^2 = 0.6097 * AB^2 + 202500
Rearranging the equation: AB^2 - 0.6097 * AB^2 = 202500
Simplifying further, we have: 0.3903 * AB^2 = 202500
Solving for AB, we get: AB ≈ √(202500 / 0.3903)
AB ≈ 640.09 ft
Therefore, the distance between the man and his friend at point A is approximately 640.09 feet.
To find the distance between the man and his friend, we need to use trigonometry and the given angles of depression.
Let's assume that the height of the hot-air balloon remains constant throughout the flight.
Let's break down the problem step-by-step:
Step 1: Identify the given information:
- The man's balloon is flying in a straight line at a constant rate of 5 feet per second.
- The angle of depression from the balloon to the friend's car is 38°.
- A minute and a half later, the angle of depression is 30°.
Step 2: Determine the height of the balloon:
Since the balloon is flying at a constant rate of 5 feet per second, after 1.5 minutes (or 90 seconds), the balloon has traveled a distance of 5 * 90 = 450 feet.
Since the height of the balloon remains constant, the distance from the balloon to the ground is also the vertical distance from the balloon to the friend's car.
Step 3: Calculate the height of the balloon:
Using the trigonometric relationship:
tan(angle of depression) = height / distance
For the first observation:
tan(38°) = height / distance
Rearranging the equation, we get:
height = distance * tan(38°)
For the second observation:
tan(30°) = height / distance
Solving for distance, we get:
distance = height / tan(30°)
Step 4: Calculate the distances using the given angles of depression:
To find the height of the balloon, we can use the angle of depression of 38° and the distance of 450 feet:
height = 450 * tan(38°)
Similarly, using the angle of depression of 30°:
distance = (450 * tan(38°)) / tan(30°)
Calculating the values, we find:
height ≈ 383.620 feet
distance ≈ 700.547 feet
Therefore, at the time when the friend is getting into the car, the distance between the man and his friend is approximately 700.547 feet.