An airplane crosses the Atlantic Ocean​ (3000 miles) with an airspeed of 500 miles per hour.  The cost C​ (in dollars) per passenger is given by c(x)=100+(x)/(25)+(34000)/(x), where x is the ground speed (airspeed and wind), what is the cost per passenger with a head wind speed of 100 miles per hour?

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.**

Learn how to calculate the cost per passenger when an airplane crosses the Atlantic Ocean against a headwind. Given an airspeed of 500 miles per hour and a headwind speed of 100 miles per hour, find the ground speed and use the cost function c(x) = 100 + x/25 + 34000/x. Suitable for advanced high school students and beyond.

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