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.
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?
1 answer
1 answer