To calculate the cost per passenger with a headwind speed of 100 miles per hour, we need to determine the ground speed of the airplane given the airspeed and the headwind.

Given:

• Airspeed of the airplane, $V_{\text{air}} = 500$ miles per hour
• Headwind speed, $W = 100$ miles per hour
• Distance across the Atlantic Ocean, $D = 3000$ miles
• Cost function per passenger, $c(x) = 100 + \frac{x}{25} + \frac{34000}{x}$, where $x$ is the ground speed

Step-by-Step Solution:

1. Calculate the Ground Speed:

   The ground speed $x$ is the airspeed minus the headwind speed: $x = V_{\text{air}} - W = 500 - 100 = 400 \text{ miles per hour}$

2. Plug Ground Speed into the Cost Function:

   Substitute $x = 400$ into the cost function $c(x)$: $c(400) = 100 + \frac{400}{25} + \frac{34000}{400}$

3. Calculate the Cost:

   $c(400) = 100 + \frac{400}{25} + \frac{34000}{400}$

   Now compute each term:

   • $\frac{400}{25} = 16$
   • $\frac{34000}{400} = 85$

   So,

   $c(400) = 100 + 16 + 85 = 201$

Conclusion:

The cost per passenger with a headwind speed of 100 miles per hour is $201.

Would you like to explore more details or have any questions?

Questions for Further Exploration:

1. How does the cost per passenger change if the headwind speed increases to 150 miles per hour?
2. What would be the cost per passenger if there were a tailwind instead of a headwind?
3. How sensitive is the cost to changes in airspeed or wind speed?
4. What happens to the cost if the distance across the Atlantic increases?
5. How would fuel efficiency changes affect the cost function?

Tip:

Understanding the impact of wind on flight costs can help optimize flight routes and reduce expenses.