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?
You must include a diagram with your answer.
Let's start by drawing a diagram to visualize the situation:
```
B
/ |
/ | d
/ |
/ θ_2|
A/______\
d/ θ_1
```
In this diagram, the balloon is represented by point A, the friend's car is represented by point B, and the distance between the two is represented by d. θ_1 and θ_2 are the angles of depression from the balloon to the friend's car at two different points in time.
From the information given, we know that the angle of depression θ_1 is 38° and the angle of depression θ_2 is 30°. We also know that the balloon is flying in a straight line at a constant rate of 5 feet per second.
To find the distance between the balloon and the friend's car, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the altitude of the balloon, which remains constant, and the adjacent side is the horizontal distance traveled by the balloon.
Let's calculate the horizontal distance traveled by the balloon from point A to point B:
Time = 1.5 minutes = 90 seconds (since the balloon flies at a constant rate of 5 feet per second)
Horizontal distance = Rate * Time = 5 feet/second * 90 seconds = 450 feet
Now we can calculate the altitude of the balloon using the tangent of the angle of depression θ_1:
tan(θ_1) = opposite/adjacent = altitude/distance
tan(38°) = altitude/450 feet
We can rearrange the equation to solve for the altitude:
altitude = tan(38°) * 450 feet
Next, we can use the altitude and the tangent of the angle of depression θ_2 to find the new distance between the balloon and the friend's car.
tan(θ_2) = opposite/adjacent = altitude/distance
tan(30°) = altitude/new distance
Again, rearranging the equation gives us:
new distance = altitude/tan(30°)
Now we can substitute the values we calculated:
altitude = tan(38°) * 450 feet
new distance = (tan(38°) * 450 feet) / tan(30°)
Calculating these values:
altitude ≈ 0.781 * 450 feet ≈ 351.45 feet
new distance ≈ (0.781 * 450 feet) / 0.577 ≈ 427.77 feet
Therefore, at that time, the distance between the man and his friend is approximately 427.77 feet.
To find the distance between the man and his friend, we can use trigonometry. Let's break down the problem into smaller steps.
Step 1: Determine the height of the balloon.
Since the balloon is flying at a constant altitude, the height remains the same throughout the observation. Let's call this height "h."
Step 2: Calculate the height difference between the first and second observation.
Between the first observation (angle of depression = 38°) and the second observation (angle of depression = 30°), the balloon has moved directly over the car. Therefore, the height difference can be calculated using trigonometry.
Let's represent the distance between the man and his friend's car as "d."
Using the tangent function:
tan(38°) = h/d₁
tan(30°) = h/d₂
where d₁ is the distance between the man and his friend's car during the first observation, and d₂ is the distance when the man is directly over the car.
Step 3: Calculate the distances d₁ and d₂.
Using the tangent function, we can rearrange the equations from Step 2 to solve for d₁ and d₂.
d₁ = h/tan(38°)
d₂ = h/tan(30°)
Step 4: Calculate the total distance traveled by the balloon.
Since the balloon is flying in a straight line at a constant rate of 5 feet per second, we can calculate the total distance traveled by multiplying the constant rate by the time in seconds.
The time is given as 1 minute and 30 seconds, which is equivalent to 90 seconds.
Total distance traveled = 5 feet/second * 90 seconds
Step 5: Calculate the remaining distance between the man and his friend's car.
To find the remaining distance between the man and his friend's car, we subtract the distance traveled (step 4) from the distance during the second observation (d₂).
Remaining distance = d₂ - (total distance traveled)
Finally, we can substitute the values into the equations and calculate the distance between the man and his friend.