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The problem in the image asks to find the average speed of a plane in still air. Here's a summary of the given information:

- The plane travels 700 miles from Los Angeles, California, to Santa Fe, New Mexico, with a wind of **32 mph**.
- The return trip (against the wind) takes **\(\frac{1}{3}\) hours** longer than the trip with the wind.

Let the speed of the plane in still air be **\( x \) mph**.

### With the wind (trip from Los Angeles to Santa Fe):
- Effective speed = \( x + 32 \) mph
- Time for this trip = \( \frac{700}{x + 32} \) hours

### Against the wind (return trip):
- Effective speed = \( x - 32 \) mph
- Time for this trip = \( \frac{700}{x - 32} \) hours

The return trip takes **\(\frac{1}{3}\) hours longer** than the initial trip:
\[
\frac{700}{x - 32} = \frac{700}{x + 32} + \frac{1}{3}
\]

### Now we can solve this equation step-by-step.

Let me calculate the value of \( x \).The solution to the equation gives two values for \( x \): **368 mph** and **-368 mph**. Since speed cannot be negative, the average speed of the plane in still air is **368 mph**.

Would you like further details or clarifications on the solution?

Here are 5 related questions to consider:
1. How would the problem change if the wind speed were higher?
2. What would be the effect if the wind direction changed midway through the flight?
3. How can we calculate the time saved when flying with a tailwind?
4. What if the plane traveled a longer distance, such as 1,000 miles?
5. Can this same method be used for a boat traveling with and against a river current?

**Tip:** When solving these types of problems, always check for physical meaning, like ensuring speeds are positive.

A plane travels 700 miles from Los Angeles, California, to Santa Fe, New Mexico, with a prevailing wind of 32 mph. The return trip against the wind takes 1/3 hour longer. Find the average speed of the plane in still air.

Learn how to find the average speed of a plane in still air, accounting for wind speed. This problem involves algebraic concepts and solving rational equations, suitable for high school students in grades 9-12. The given data includes a 700-mile journey with a wind speed of 32 mph and a time difference of 1/3 hour.

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