Calculating Average Speed for the Entire Journey

This problem asks us to find the average speed for a journey that consists of three different parts, each with a specific distance and speed.

Understanding Average Speed

The average speed is not simply the average of the speeds for each part. Instead, it's calculated by dividing the total distance traveled by the total time taken for the entire journey.

The formula is:

$ Average Speed = \frac{Total Distance}{Total Time} $

Breaking Down the Journey

The journey has three segments:

• Segment 1: Distance = 488 km, Speed = 61 km/hr
• Segment 2: Distance = 468 km, Speed = 78 km/hr
• Segment 3: Distance = 210 km, Speed = 35 km/hr

Calculating Time for Each Segment

We need to calculate the time taken for each part of the journey using the formula: $ Time = \frac{Distance}{Speed} $.

• Time for Segment 1:

  $ T_1 = \frac{488 \text{ km}}{61 \text{ km/hr}} = 8 \text{ hours} $

• Time for Segment 2:

  $ T_2 = \frac{468 \text{ km}}{78 \text{ km/hr}} = 6 \text{ hours} $

• Time for Segment 3:

  $ T_3 = \frac{210 \text{ km}}{35 \text{ km/hr}} = 6 \text{ hours} $

Calculating Total Distance and Total Time

Now, let's find the total distance and total time for the whole trip.

• Total Distance:

  $ Total Distance = 488 \text{ km} + 468 \text{ km} + 210 \text{ km} = 1166 \text{ km} $

• Total Time:

  $ Total Time = T_1 + T_2 + T_3 = 8 \text{ hours} + 6 \text{ hours} + 6 \text{ hours} = 20 \text{ hours} $

Calculating the Average Speed

Finally, we can calculate the average speed using the total distance and total time.

$ Average Speed = \frac{Total Distance}{Total Time} = \frac{1166 \text{ km}}{20 \text{ hours}} $

$ Average Speed = 58.3 \text{ km/hr} $

Conclusion

The average speed for the woman's entire journey is 58.3 km/hr.