Questions: An airplane crosses the Atlantic Ocean (3000 miles) with an airspeed of 550 miles per hour. The cost C (in dollars) per passenger is given by C(x) = 75 + x/10 + 35,000/x where x is the ground speed (airspeed ± wind). 193.64 (Round to the nearest cent as needed.) (b) What is the cost per passenger with a head wind of 50 miles per hour? 195 (Round to the nearest cent as needed.) (c) What is the cost per passenger with a tail wind of 100 miles per hour? 193.85 (Round to the nearest cent as needed.) (d) What is the cost per passenger with a head wind of 100 miles per hour? (Round to the nearest cent as needed.)

Solution Steps

To solve the given problem, we need to calculate the cost per passenger using the provided cost function \( C(x) = 75 + \frac{x}{10} + \frac{35,000}{x} \). The ground speed \( x \) is determined by adjusting the airspeed of 550 miles per hour with the wind speed. For each scenario, we will substitute the appropriate ground speed into the cost function and compute the result.

1. Head Wind of 50 miles per hour: The ground speed is the airspeed minus the wind speed, i.e., \( x = 550 - 50 \).
2. Tail Wind of 100 miles per hour: The ground speed is the airspeed plus the wind speed, i.e., \( x = 550 + 100 \).
3. Head Wind of 100 miles per hour: The ground speed is the airspeed minus the wind speed, i.e., \( x = 550 - 100 \).

To find the ground speed with a head wind of 50 mph, subtract the wind speed from the airspeed: \[ x = 550 - 50 = 500 \text{ mph} \]

Substitute \( x = 500 \) into the cost function: \[ C(500) = 75 + \frac{500}{10} + \frac{35,000}{500} \] \[ C(500) = 75 + 50 + 70 = 195.0 \]

To find the ground speed with a tail wind of 100 mph, add the wind speed to the airspeed: \[ x = 550 + 100 = 650 \text{ mph} \]

Substitute \( x = 650 \) into the cost function: \[ C(650) = 75 + \frac{650}{10} + \frac{35,000}{650} \] \[ C(650) = 75 + 65 + 53.8462 \approx 193.85 \]

To find the ground speed with a head wind of 100 mph, subtract the wind speed from the airspeed: \[ x = 550 - 100 = 450 \text{ mph} \]

Substitute \( x = 450 \) into the cost function: \[ C(450) = 75 + \frac{450}{10} + \frac{35,000}{450} \] \[ C(450) = 75 + 45 + 77.7778 \approx 197.78 \]

Final Answer