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 180 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?
Well, Carissa seems to have quite the range of speeds! Let's see if we can calculate her average speed during the round trip.
To start, we need to figure out the distances Carissa walked uphill, across flat ground, and downhill. Let's call these distances d1, d2, and d3, respectively.
Now, we know that Carissa walked at different speeds for each segment of her journey. Since speed = distance/time, we can find the time it took for each segment:
Time for uphill = d1/100
Time for flat = d2/120
Time for downhill = d3/180
Since Carissa took the same route in reverse to return to the café, the total time for the round trip is:
Total time = Time for uphill + Time for flat + Time for downhill
Now, we can calculate Carissa's average speed:
Average speed = Total distance (2 * (d1 + d2 + d3)) / Total time
So, her average speed for the entire round trip would be:
Average speed = 2 * (d1 + d2 + d3) / (d1/100 + d2/120 + d3/180)
Now that we have the formula, all we need are the actual distances d1, d2, and d3 to crunch the numbers and find Carissa's average speed.
To find Carissa's average speed during the entire round trip, we need to calculate the total distance she traveled and divide it by the total time taken.
Let's assume the distance between the café and the boba shop is "x" meters.
Since Carissa walks in the same route back and forth, the total distance traveled is:
Total distance = 2 * x meters
Next, we need to calculate the time taken for each leg of the journey. Let:
- Uphill speed = 100 meters per hour
- Flat ground speed = 120 meters per hour
- Downhill speed = 180 meters per hour
The time taken to go from the café to the boba shop is:
Time taken uphill = x / Uphill speed
The time taken to return from the boba shop to the café is:
Time taken downhill = x / Downhill speed
The time taken on flat ground is the same for both leg, so let's call it:
Time taken flat ground = t
The total time taken for the entire round trip is:
Total time = Time taken uphill + Time taken flat ground + Time taken downhill
Total time = x / 100 + t + x / 180
Now, we can find the average speed:
Average speed = Total distance / Total time
Average speed = 2 * x / (x / 100 + t + x / 180)
Simplifying the equation:
Average speed = 2 * 100 * 180 * x / (180x + 100xt + 100xt)
Average speed = 36000 / (360 + 2t)
Hence, Carissa's average speed during the entire round trip is 36000 / (360 + 2t).
To find Carissa's average speed during the entire round trip, we need to calculate the total distance she walked and the total time it took her.
First, let's break down the different segments of the trip:
1. Walking from the café to the boba shop.
2. Walking from the boba shop back to the café.
Since Carissa takes the same route in reverse, the distance covered in both segments is the same. Let's denote this distance as d.
Now, let's calculate the time it took Carissa to complete each segment:
1. Walking from the café to the boba shop:
Distance = d (same as the return trip)
Speed = 120 meters per hour
Time taken = Distance / Speed = d / 120
2. Walking from the boba shop back to the café:
Distance = d (same as the outgoing trip)
Speed = average of uphill speed (100), flat ground speed (120), and downhill speed (180)
= (100 + 120 + 180) / 3 = 400 / 3
Time taken = Distance / Speed = d / (400 / 3)
Now, let's calculate the total time taken for the round trip:
Total time taken = Time taken for the outgoing trip + Time taken for the return trip
= d / 120 + d / (400 / 3)
= 3d / 400 + 3d / 400
= 6d / 400
= 3d / 200
Finally, the average speed during the entire round trip is given by:
Average speed = Total distance covered / Total time taken
= 2d / (3d / 200)
= 400 / 3
Therefore, Carissa's average speed during the entire round trip is 400 / 3 meters per hour.