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 \( C(x)=75+\frac{x}{10}+\frac{32,000}{x} \) where \( x \) is the ground speed (airspeed \( \pm \) wind). (a) What is the cost per passenger for quiescent (no wind) conditions? \( \$ 188.18 \) (Round to the nearest cent as needed.) (b) What is the cost per passenger with a head wind of 50 miles per hour? \( \$ 189 \) (Round to the nearest cent as needed.) (c) What is the cost per passenger with a tail wind of 100 miles per hour? \( \$ \square \) (Round to the nearest cent as needed.)

To solve the problem, we will follow these steps: 1. **Identify the known conditions:** - Airspeed of the airplane: \( 550 \) miles per hour. - Distance across the Atlantic Ocean: \( 3000 \) miles. - Cost function: \( C(x) = 75 + \frac{x}{10} + \frac{32,000}{x} \). 2. **Calculate the ground speed for each scenario:** - For quiescent conditions (no wind), the ground speed \( x \) is equal to the airspeed. - For a headwind of \( 50 \) miles per hour, the ground speed \( x \) is the airspeed minus the wind speed. - For a tailwind of \( 100 \) miles per hour, the ground speed \( x \) is the airspeed plus the wind speed. 3. **Substitute the ground speed into the cost function to find the cost per passenger.** ### (a) Cost per passenger for quiescent conditions (no wind) - Ground speed \( x = 550 \) mph. - Substitute \( x \) into the cost function: \[ C(550) = 75 + \frac{550}{10} + \frac{32,000}{550} \] Calculating each term: 1. \( \frac{550}{10} = 55 \) 2. \( \frac{32,000}{550} \approx 58.18 \) Now, summing these values: \[ C(550) = 75 + 55 + 58.18 = 188.18 \] Thus, the cost per passenger for quiescent conditions is \( \$188.18 \). ### (b) Cost per passenger with a headwind of 50 miles per hour - Ground speed \( x = 550 - 50 = 500 \) mph. - Substitute \( x \) into the cost function: \[ C(500) = 75 + \frac{500}{10} + \frac{32,000}{500} \] Calculating each term: 1. \( \frac{500}{10} = 50 \) 2. \( \frac{32,000}{500} = 64 \) Now, summing these values: \[ C(500) = 75 + 50 + 64 = 189 \] Thus, the cost per passenger with a headwind of 50 miles per hour is \( \$189 \). ### (c) Cost per passenger with a tailwind of 100 miles per hour - Ground speed \( x = 550 + 100 = 650 \) mph. - Substitute \( x \) into the cost function: \[ C(650) = 75 + \frac{650}{10} + \frac{32,000}{650} \] Calculating each term: 1. \( \frac{650}{10} = 65 \) 2. \( \frac{32,000}{650} \approx 49.23 \) Now, summing these values: \[ C(650) = 75 + 65 + 49.23 = 189.23 \] Thus, the cost per passenger with a tailwind of 100 miles per hour is \( \$189.23 \). ### Final Answers: - (a) \( \$188.18 \) - (b) \( \$189 \) - (c) \( \$189.23 \)