Not counting the "turn around" , a helicopter carrying passengers between an airport and the roof of a downtown skycraper 20 miles away can make the round trip in 15 minutes. The helicopter flies 2 miles away with the wind when the time it flies 1 mile against the wind. What is the wind speed?
To solve this problem, we need to break it down into smaller steps. Let's start by understanding the given information:
1. The round trip between the airport and the downtown skyscraper takes 15 minutes.
2. The distance between the airport and the skyscraper is 20 miles.
3. The helicopter flies 2 miles with the wind in the same time it flies 1 mile against the wind.
Now, let's calculate the helicopter's speed in still air. We can assume that the helicopter's speed is constant during the entire trip.
Let's say the helicopter's speed in still air is "S" miles per minute, and the wind speed is "W" miles per minute.
On the way to the skyscraper (downwind):
- Distance: 20 miles
- Helicopter's speed: S + W (with the wind)
- Time taken: (20 miles) / (S + W)
On the way back to the airport (against the wind):
- Distance: 20 miles
- Helicopter's speed: S - W (against the wind)
- Time taken: (20 miles) / (S - W)
Given that the total round trip time is 15 minutes, we can set up the equation:
(20 miles) / (S + W) + (20 miles) / (S - W) = 15 minutes
Now, to find the wind speed (W), we need to solve this equation.
To simplify the equation, we can multiply both sides by (S + W) and (S - W) to eliminate the denominators:
20(S - W) + 20(S + W) = 15(S + W)(S - W)
Now we can solve for W:
40S = 15(S^2 - W^2)
40S = 15S^2 - 15W^2
15W^2 + 40S - 15S^2 = 0
To solve this quadratic equation, we can use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 15, b = 40, and c = 0. Plugging these values into the formula gives us:
W = (-40 ± √(40^2 - 4 * 15 * 0)) / (2 * 15)
Simplifying further:
W = (-40 ± √(1600)) / 30
W = (-40 ± 40) / 30
Now, we have two possible solutions:
1. W = (-40 - 40) / 30 = -80 / 30 = -2.67 miles per minute
2. W = (-40 + 40) / 30 = 0 miles per minute
Since wind speed cannot be negative, the only valid solution is W = 0 miles per minute. This means there is no wind.
Therefore, the wind speed in this scenario is 0 miles per minute.