An airplane crosses the Atlantic Ocean​ (3000 miles) with an airspeed of 600 miles per hour.  The cost C​ (in dollars) per passenger is given by
​C(x)equals150plusStartFraction x Over 15 EndFraction
plusStartFraction 34 comma 000 Over x EndFraction
​,
where x is the ground speed​ (airspeed plus or minus ​wind).

The cost function for the airplane flight per passenger is given as:

\[
C(x) = 150 + \frac{x}{15} + \frac{34,000}{x}
\]

where:
- \( x \) is the ground speed, which is the airspeed plus or minus the wind speed (measured in miles per hour),
- 150 is a fixed cost,
- \( \frac{x}{15} \) is a cost component that increases with ground speed, and
- \( \frac{34,000}{x} \) is a cost component that decreases as the ground speed increases.

### Further Exploration
1. How would the cost function change if the airspeed was different?
2. At what ground speed \( x \) is the cost per passenger minimized?
3. How does wind speed affect the ground speed and thus the cost?
4. How would the cost function change if the distance traveled was not 3,000 miles but a different value?
5. Can we find the derivative of the cost function to analyze how \( C(x) \) changes with respect to \( x \)?

#### Tip:
Ground speed is influenced by both the airspeed of the plane and the wind conditions, which can either increase or decrease ground speed depending on whether it's a tailwind or headwind.

An airplane crosses the Atlantic Ocean (3000 miles) with an airspeed of 600 miles per hour. The cost C (in dollars) per passenger is given by C(x) = 150 + x/15 + 34,000/x, where x is the ground speed (airspeed plus or minus wind).

Explore how to optimize the cost function C(x) = 150 + x/15 + 34,000/x for airplane travel over 3000 miles. This solution covers key mathematical concepts including rational expressions, optimization, and derivative analysis to find the minimal cost per passenger. Suitable for high school students in Grades 10-12.

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