A traveler flies from Charlotte, NC to Los Angeles, CA. At the same time, another traveler flies from Los Angeles to Charlotte. The air speed of each plane is the same. The ground... A traveler flies from Charlotte, NC to Los Angeles, CA. At the same time, another traveler flies from Los Angeles to Charlotte. The air speed of each plane is the same. The ground speeds are shown below. What is the air speed? What is the wind speed? CA to NC Ground Speed= 550mi/h Air Speed = a Wind Speed = w NC to CA Ground Speed = 495 mi/h Air Speed = a
Understand the Problem
The problem describes two planes flying between Charlotte, NC and Los Angeles, CA. One plane flies from CA to NC with a ground speed of 550 mi/h, while the other flies from NC to CA with a ground speed of 495 mi/h. Both planes have the same air speed, denoted as 'a', and the wind speed is denoted 'w'. The task is to determine the air speed 'a' and the wind speed 'w'.
Answer
Air speed is $522.5$ mi/h and wind speed is $27.5$ mi/h.
Answer for screen readers
Air speed, $a = 522.5$ mi/h Wind speed, $w = 27.5$ mi/h
Steps to Solve
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Set up equations for ground speed in terms of air speed and wind speed. For the plane flying from CA to NC, the ground speed is the air speed plus the wind speed: $550 = a + w$ For the plane flying from NC to CA, the ground speed is the air speed minus the wind speed: $495 = a - w$
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Solve the system of equations for $a$ and $w$. We can use the method of elimination to solve this system. Add the two equations together: $550 + 495 = (a + w) + (a - w)$ $1045 = 2a$
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Solve for $a$. Divide both sides by 2: $a = \frac{1045}{2} = 522.5$
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Substitute the value of $a$ back into one of the original equations to solve for $w$. Using the first equation: $550 = 522.5 + w$
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Solve for $w$. Subtract 522.5 from both sides: $w = 550 - 522.5 = 27.5$
Air speed, $a = 522.5$ mi/h Wind speed, $w = 27.5$ mi/h
More Information
The air speed is the speed of the plane relative to the air. The wind either increases or decreases the ground speed which is the actual speed of the plane relative to the ground.
Tips
A common mistake would be to subtract the two original equations instead of adding them. This would eliminate $a$ instead of $w$ and lead to solving for the wind speed first, which still works but can be confusing.
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