Question
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?
Answers
GPT 3.5
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.
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.