Carissa the chessplayer is a very, very slow walker. In fact, she walks at $100$ meters per hour when walking uphill, $120$ meters per hour when walking across flat ground, and $150$ meters per hour when walking downhill. One day, Carissa walks across Boston from a café to a boba shop, and then takes the same route in reverse to return to the café. What was Carissa's average speed during the entire round trip?

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To determine Carissa's average speed during her round trip from the café to the boba shop and back, we must first recognize that her speed varies depending on whether she is walking uphill, on flat ground, or downhill. Suppose the distance she walks uphill is $ d_u $, the distance on flat ground is $ d_f $, and the distance downhill is $ d_d $. The times taken for each segment are given by the distance divided by the respective speeds: $ t_u = \frac{d_u}{100} $, $ t_f = \frac{d_f}{120} $, and $ t_d = \frac{d_d}{150} $. The total distance of the one-way trip is $ d_u + d_f + d_d $, and she takes the same route back, making the total distance of the round trip $ 2(d_u + d_f + d_d) $.

Next, we calculate the total time taken for the round trip. The time for the one-way trip is the sum of the times for each segment:
\[ t_{\text{one-way}} = t_u + t_f + t_d = \frac{d_u}{100} + \frac{d_f}{120} + \frac{d_d}{150} \]
Since she walks the same route back, the time for the round trip is:
\[ t_{\text{round trip}} = 2 \left( \frac{d_u}{100} + \frac{d_f}{120} + \frac{d_d}{150} \right) \]

The average speed is defined as the total distance divided by the total time. Therefore, the average speed $ v_{\text{avg}} $ is:
\[ v_{\text{avg}} = \frac{\text{total distance}}{\text{total time}} = \frac{2(d_u + d_f + d_d)}{2 \left( \frac{d_u}{100} + \frac{d_f}{120} + \frac{d_d}{150} \right)} = \frac{d_u + d_f + d_d}{\frac{d_u}{100} + \frac{d_f}{120} + \frac{d_d}{150}} \]

This formula consolidates the distances and speeds into one average speed. By using the given speeds and distances, we can compute the exact average speed. In summary, Carissa's average speed for the entire round trip can be calculated using the formula above.

Do you want more details or have any questions about this problem? 

Here are some related questions to expand your understanding:

1. What is the significance of using harmonic mean in calculating average speed for different segments?
2. How does the average speed change if Carissa's downhill speed increases?
3. What happens to the average speed if the distance of flat ground is doubled?
4. How would you calculate the average speed if there were additional segments with different speeds?
5. Can you derive a general formula for average speed given any number of segments with varying speeds?
6. How would her average speed be affected if she walked faster uphill than on flat ground?
7. What is the average speed if Carissa takes a different return route with different distances and speeds?
8. How do you apply the concept of average speed to other types of motion, such as driving a car on varying terrain?

**Tip:** When calculating average speeds over different segments, remember to account for the different times spent in each segment rather than simply averaging the speeds directly.

Learn how to calculate Carissa's average speed during her round trip from a café to a boba shop and back in Boston. This problem involves speed, distance, and time calculations across different terrains. Suitable for high school students.

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