Question

John flies from Atlanta, Georgia to San Francisco, California. It takes 5.6 hours to travel 2100 miles against the head wind. At the same time,Debby flies from San Francisco to Atlanta. Her plane travels with the same average airspeed, but the tail wind, and her flight takes only 4.8 hours. Write a system of equation that relate time, airspeed, and wind speed to distance for each traveler. Solve the system to fine the air speed

Answer 1

The average airspeed of the two planes is
`406.25\; \rm mph`.

Explanation:

Let
`x` denotes the average airspeed (in miles-per-hour) of the two planes. Let
`y` denote the average speed of wind (also in miles-per-hour) along that route.

John is travelling against the head wind. Therefore, his ground speed would be the difference between his airspeed and the speed of the wind. That is:

`\begin{aligned} & v(\text{John, ground speed})\\ =& v(\text{John, airspeed}) - v(\text{wind speed}) \\=& x - y\end{aligned}`.

On the other hand, Debby is travelling in the tail wind. Assume that Debby and John are taking the same route but in the opposite directions. The ground speed of Debby would be the sum of her airspeed and the speed of the wind:

`\begin{aligned} & v(\text{Debby, ground speed})\\ =& v(\text{Debby, airspeed}) + v(\text{wind speed}) \\=& x + y\end{aligned}`.

Keep in mind that:

`\text{Distance Traveled} = \text{Average Speed} * \text{Time}`.

This equation can help relate time and ground speed to distance.

John traveled
`2100` miles in
`5.6` hours at a ground speed of
`(x - y) \; \rm mph`. Therefore:

`5.6\; (x - y) = 2100`.

Similarly, Debby traveled
`2100` miles in
`4.8` hours at a ground speed of
`(x + y)\; \rm mph`. Therefore:

`4.8\; (x + y) = 2100`.

Combine these two equations to obtain:

`\left\lbrace\begin{aligned}& 5.6\; (x - y) = 2100 \\ & 4.8\; (x + y) = 2100\end{aligned}\right.`.

Solve this system of equations for
`x` and
`y`:

`\left\lbrace\begin{aligned}&x = 406.25 \\ &y = 31.25\end{aligned}\right.`.

In other words:

• the average airspeed of the two aircrafts is
  `406.25\; \rm mph`, while
• the average wind speed along that route is
  `31.25\; \rm mph`.