Two boats start together and race across a 42 km wide lake and back. Boat A goes across at 42 km/h and returns at 42 km/h. Boat B goes across at 21 km/h, and its crew, realizing how far behind it is getting, returns at 63 km/h. Turnaround times are negligible, and the boat that completes the round-trip first wins.
(a) Which boat wins? (Or is it a tie?)
boat A
boat B
it's a tie
By how much?
km
(b) What is the average velocity of the winning boat?
km/h
I have no idea h
To determine which boat wins the race, we need to find the total time it takes for each boat to complete the round trip.
Boat A:
Since the boats start together, it takes Boat A the same amount of time to cross as it does to return. The distance for each leg of the trip is 42 km, and the speed is 42 km/h. So, the time it takes for Boat A to complete the round trip is:
Time_A = (Distance_Across / Speed_Across) + (Distance_Back / Speed_Back)
= (42 km / 42 km/h) + (42 km / 42 km/h)
= 2 hours
Boat B:
For Boat B, its speed is different for each leg of the trip. The time it takes for Boat B to complete the round trip can be calculated as follows:
Time_B = (Distance_Across / Speed_Across) + (Distance_Back / Speed_Back)
= (42 km / 21 km/h) + (42 km / 63 km/h)
= 2 + 2/3 hours
= 2.67 hours (rounded to two decimal places)
Now that we know the total time for each boat, we compare the times:
Time_A = 2 hours
Time_B = 2.67 hours
Since Boat A's total time is shorter than Boat B's time, Boat A wins the race.
To find out the distance by which Boat A wins, we subtract the time it takes for Boat B from the time it takes for Boat A and multiply it by Boat A's speed:
Distance_Winner = (Time_A - Time_B) * Speed_Across
= (2 hours - 2.67 hours) * 42 km/h
= -0.67 hours * 42 km/h
= -28.14 km/h (negative value indicates Boat A wins by this distance)
Hence, Boat A wins the race by 28.14 km (approximately).
Finally, to calculate the average velocity of the winning boat (Boat A), we divide the total distance it travels in the round trip by the total time it takes:
Average Velocity_A = Total Distance_A / Total Time_A
= (42 km + 42 km) / 2 hours
= 84 km / 2 hours
= 42 km/h
Therefore, the average velocity of the winning boat (Boat A) is 42 km/h.