https://r2.erweima.ai/mathbot/image/1724164533782707w5uah.jpg

Let's analyze the problem in the image to find the value of \( x \).

### Given:
1. Jo drives her car for \( x \) minutes at a speed of \( 48 \, \text{km/h} \).
2. She then continues along \( 15 \, \text{km} \) in \( 20 \) minutes.
3. The average speed for the entire journey is \( 41.4 \, \text{km/h} \).

### Steps to Find \( x \):

1. **Convert the time into hours** for consistency:
   - \( x \) minutes is \( \frac{x}{60} \) hours.
   - \( 20 \) minutes is \( \frac{20}{60} = \frac{1}{3} \) hours.

2. **Calculate the distance covered in the first part** of the journey:
   - Distance = Speed \(\times\) Time
   - Distance for first part \( = 48 \times \frac{x}{60} = \frac{48x}{60} = \frac{4x}{5} \, \text{km} \).

3. **Total distance covered** in the entire journey:
   - Total distance = Distance for first part \( + \) Distance for second part
   - \( \text{Total distance} = \frac{4x}{5} + 15 \, \text{km} \).

4. **Total time taken**:
   - Time for first part \( = \frac{x}{60} \) hours.
   - Time for second part \( = \frac{1}{3} \) hours.
   - Total time \( = \frac{x}{60} + \frac{1}{3} \) hours.

5. **Use the formula for average speed**:
   \[
   \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}
   \]
   Given that the average speed is \( 41.4 \, \text{km/h} \), we set up the equation:
   \[
   41.4 = \frac{\frac{4x}{5} + 15}{\frac{x}{60} + \frac{1}{3}}
   \]

6. **Simplify and solve for \( x \)**:

   Multiply both sides by the denominator:
   \[
   41.4 \times \left(\frac{x}{60} + \frac{1}{3}\right) = \frac{4x}{5} + 15
   \]

   Distribute \( 41.4 \):
   \[
   \frac{41.4x}{60} + 41.4 \times \frac{1}{3} = \frac{4x}{5} + 15
   \]
   Simplify the terms:
   \[
   \frac{41.4x}{60} + 13.8 = \frac{4x}{5} + 15
   \]
   Convert to a common denominator to solve for \( x \).

7. **Further simplification**:
   Multiply the entire equation by 60 to eliminate fractions:
   \[
   41.4x + 828 = 48x + 900
   \]
   Solve for \( x \):
   \[
   828 - 900 = 48x - 41.4x
   \]
   Simplifying:
   \[
   -72 = 6.6x
   \]
   \[
   x = \frac{-72}{6.6} = -10.91
   \]

This result suggests that there may be a mistake in the calculations or problem setup. Normally, \( x \) should be positive and realistic. To correct this, re-check the steps or constraints. 

Would you like to go through the calculations again or clarify any details? 

---

### 8 Relative Questions:
1. What assumptions do we make when solving problems involving average speed?
2. How would the result change if the average speed was 50 km/h instead of 41.4 km/h?
3. Why is it important to convert all time units to hours in speed problems?
4. What other information would be needed if we knew only the total time but not the distance?
5. How does the average speed formula differ from the instantaneous speed formula?
6. What real-world factors could affect the accuracy of these types of calculations?
7. How would the problem change if the second part of the journey had a different speed?
8. How can we verify the solution to such problems?

### Tip:
When working with equations involving speed, time, and distance, ensure all units are consistent (typically converting everything to hours and kilometers).

Learn how to find the value of x in a speed, time, and distance problem involving different speeds and distances. Detailed step-by-step solution provided with calculations and explanations. Suitable for high school students.

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