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?
thx for taking the time to do this.
To calculate Carissa's average speed during the entire round trip, we need to first determine the total distance she walked and the time it took her to complete the trip.
Let's assume that the distance between the café and boba shop is x kilometers. When walking uphill, Carissa's speed is 100 meters per hour, so it would take her x/0.1 = 10x hours to cover this distance. When walking across flat ground, her speed is 120 meters per hour, so she would cover the same distance in x/0.12 = 10/1.2 x = 25/3 x hours. When walking downhill, her speed is 150 meters per hour, so she would cover the same distance in x/0.15 = 20/3 x hours.
Thus, the total time it took Carissa to walk from the café to the boba shop was 10x + 25/3 x + 20/3 x = 100/3 x hours. When she walked back from the boba shop to the café, she covered the same distance, so the total time for the round trip was 200/3 x hours.
The total distance of the round trip was 2x kilometers. Therefore, Carissa's average speed during the entire round trip was:
average speed = total distance / total time = 2x / (200/3 x) = 6/5 x/hour = 1.2x kilometers per hour.
So Carissa's average speed during the entire round trip was 1.2x kilometers per hour.
The route may be hilly, up then down then up then level then down again, and so on.
Suppose that the total distance uphill is U, the the total distance on level ground is L and the total downhill is D.
The time taken for the outward trip T1 would be equal to U/100 + L/120 + D/150.
On the return journey, uphill becomes downhill and downhill becomes uphill, level stays the same.
The time taken for the return trip T2 will then be U/150 + L/120 + D/100.
The total time for the round trip will be T1 + T2 = U/60 + L/60 + D/60 = ( U + L +D)/60
The total distance TD = 2(U + L + D), so the average speed will be TD/(T1 + T2) = 120 km/ hr.
We can use the formula for average speed, which is the total distance traveled divided by the total time taken. Let's call the total distance traveled "d".
Since Carissa took the same route both ways, the distance traveled uphill is the same in both directions, and the same is true for the distance traveled downhill and across flat ground. So, the total distance traveled uphill is 2d/3, the total distance traveled across flat ground is d/3, and the total distance traveled downhill is 2d/3.
The time taken to travel uphill at 100 meters per hour is 2d/3 * (1 hour/100 meters) = 2d/(300 meters), the time taken to travel across flat ground at 120 meters per hour is d/3 * (1 hour/120 meters) = d/(360 meters), and the time taken to travel downhill at 180 meters per hour is 2d/3 * (1 hour/180 meters) = 2d/540 meters.
The total time taken is 2d/(300 meters) + d/(360 meters) + 2d/540 meters = (12d + 12d + 10d)/(180 meters) = 24d/180 meters = 4d/30 meters.
The average speed during the entire round trip is d/(4d/30 meters) = 30/4 = 7.5 meters per hour.
