https://file.aiquickdraw.com/mathbot/file/1726447531010xkljtdw7.jpg

Let's solve the problem step by step based on the information provided in the image.

The cost per passenger is given by the function:

\[
C(x) = 100 + \frac{x}{7} + \frac{40,000}{x}
\]

where:
- \( x \) is the ground speed (airspeed ± wind) in miles per hour,
- 3000 miles is the distance traveled.

### (a) Cost at ground speeds of 430 mph and 560 mph:
We will calculate \( C(x) \) for \( x = 430 \) and \( x = 560 \).

The formula is:
\[
C(x) = 100 + \frac{x}{7} + \frac{40,000}{x}
\]

1. For \( x = 430 \):
\[
C(430) = 100 + \frac{430}{7} + \frac{40,000}{430}
\]

2. For \( x = 560 \):
\[
C(560) = 100 + \frac{560}{7} + \frac{40,000}{560}
\]

### (b) Domain of \( C(x) \):
- \( x \) represents the ground speed. Since \( x \) is a real and positive value (ground speed cannot be zero or negative), the domain of \( C(x) \) is \( x > 0 \).

### (c) Graph of \( C(x) \):
To graph \( C(x) \), it would be helpful to plot the function over a range of \( x \) values using a graphing calculator or software. From the problem, you can see how \( C(x) \) behaves as \( x \) increases.

### (d) Table with TblStart = 0 and ∆Tbl = 50:
The table shown in the image contains values of \( C(x) \) at specific \( x \) values, such as 450, 500, 550, and 600. We could generate the values of \( C(x) \) for these values and others.

### (e) Ground speed that minimizes cost:
The table suggests the minimum cost occurs between 500 and 550 mph. To the nearest 50 mph, we could use either 500 or 550 mph as the speed that minimizes the cost.

Let's compute parts (a) and (e) first. I'll calculate the costs for the specified speeds.The calculated costs are:

- \( C(430) \approx 254.45 \) dollars per passenger.
- \( C(560) \approx 251.43 \) dollars per passenger.

### (e) To minimize cost:
From the table and the calculation so far, the cost per passenger decreases as the ground speed increases from 430 mph to 560 mph. Based on the table provided in the image, the ground speed that minimizes the cost is approximately **550 miles per hour**.

Would you like further details or help with the graph, table, or domain? Here are 5 related questions to expand your understanding:

1. How does the cost function behave as the ground speed increases further beyond 600 mph?
2. What are the implications of the terms \( \frac{x}{7} \) and \( \frac{40,000}{x} \) in the cost function?
3. How can we determine the exact minimum cost using calculus (derivatives)?
4. What impact would a change in the distance (3000 miles) have on the cost function?
5. How could you estimate the cost per passenger if the ground speed varies due to wind conditions?

**Tip:** Minimizing the cost function often involves checking both the derivative and boundary behavior of the function.

A plane 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/7 + 40,000/x, where x is the ground speed (airspeed ± wind). What is the cost when the ground speed is 430 miles per hour; 560 miles per hour? Find the domain of C. Use a graphing calculator to graph the function C = C(x). To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Learn how to calculate and minimize the cost per passenger for a transatlantic flight based on ground speed. This example covers algebraic manipulation, cost function graphing, and optimization techniques. Suitable for high school students in Grades 10-12.

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