a flight to and from is 3000 m and takes 7 hours. the plane travels east and west, with and against the wind. the speed in still air is 450 km/h. what is the speed of the wind?

Let the speed of the wind be \( w \) km/h.

### Given information:
- The total round trip distance is 3000 km (1500 km each way).
- Total time taken is 7 hours.
- The plane's speed in still air is 450 km/h.

### When traveling east (with the wind):
- The plane's effective speed is \( 450 + w \) km/h.

### When traveling west (against the wind):
- The plane's effective speed is \( 450 - w \) km/h.

### Time for each leg of the journey:
- Time to travel east (with the wind) is \( \frac{1500}{450 + w} \) hours.
- Time to travel west (against the wind) is \( \frac{1500}{450 - w} \) hours.

### Total time:
The total time for both legs of the journey is 7 hours:
\[
\frac{1500}{450 + w} + \frac{1500}{450 - w} = 7
\]

Now, solve this equation to find \( w \).

#### Step 1: Multiply both sides by \( (450 + w)(450 - w) \) to eliminate the denominators:
\[
1500(450 - w) + 1500(450 + w) = 7(450 + w)(450 - w)
\]

#### Step 2: Simplify both sides:
\[
1500(450 - w) + 1500(450 + w) = 7(450^2 - w^2)
\]
\[
1500 \cdot 450 - 1500 \cdot w + 1500 \cdot 450 + 1500 \cdot w = 7(202500 - w^2)
\]
\[
3000 \cdot 450 = 7(202500 - w^2)
\]

#### Step 3: Simplify further:
\[
1350000 = 7(202500 - w^2)
\]
\[
1350000 = 1417500 - 7w^2
\]

#### Step 4: Solve for \( w^2 \):
\[
1350000 - 1417500 = -7w^2
\]
\[
-67500 = -7w^2
\]
\[
w^2 = \frac{67500}{7}
\]
\[
w^2 = 9642.86
\]

#### Step 5: Take the square root of both sides:
\[
w = \sqrt{9642.86} \approx 98.21
\]

### Conclusion:
The speed of the wind is approximately **98.21 km/h**.

Would you like more details on the solution or any further clarification?

### Related Questions:
1. How does the plane's total time change if the wind speed increases by 10 km/h?
2. If the wind speed is reduced by half, what will be the new total time for the trip?
3. What would be the plane's effective speed traveling west with a 120 km/h wind speed?
4. How would the total travel time change if the distance were increased by 500 km each way?
5. What would the wind speed be if the round trip took 8 hours instead of 7?

### Tip:
When solving problems involving relative motion, it's important to carefully distinguish between motion with the wind (added speed) and against the wind (subtracted speed).

A flight to and from is 3000 m and takes 7 hours. The plane travels east and west, with and against the wind. The speed in still air is 450 km/h. What is the speed of the wind?

Solve a problem involving a round trip flight of 3000 km with a total time of 7 hours. Learn how to calculate the speed of the wind using algebraic methods and quadratic equations, where the plane's speed in still air is 450 km/h. This problem is suitable for students in Grades 9-12.

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