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 how to do th
To solve this problem, we need to calculate the time taken by each boat to complete the round trip and compare them to determine the winner.
Let's begin by calculating the time taken by Boat A. Since the distance across the lake and back is 42 km each way, Boat A's speed is 42 km/h. Therefore, the time taken by Boat A to go across and come back is:
Time taken by Boat A = (Distance / Speed) = (42 km / 42 km/h) = 1 hour
Now, let's determine the time taken by Boat B. Boat B's speed when going across the lake is 21 km/h, so the time taken for the forward journey is:
Time taken by Boat B (forward journey) = (Distance / Speed) = (42 km / 21 km/h) = 2 hours
For the return trip, Boat B's speed is 63 km/h, so the time taken is:
Time taken by Boat B (return journey) = (Distance / Speed) = (42 km / 63 km/h) = 2/3 hours
Adding the forward and return journey times, we find the total time taken by Boat B:
Total time taken by Boat B = 2 hours + 2/3 hours = 8/3 hours
Comparing the times, we see that Boat A takes 1 hour to complete the round trip, while Boat B takes 8/3 hours. Therefore, Boat A is the winner.
To calculate the winning margin in kilometers, we subtract the distance covered by Boat B from the distance covered by Boat A:
Winning margin = Distance covered by Boat A - Distance covered by Boat B
= 42 km - 42 km
= 0 km
Hence, the answer to part (a) is that Boat A wins, and there is no winning margin as it is a tie.
For part (b), since Boat A is the winner, the average velocity of the winning boat is the total distance covered divided by the total time taken:
Average velocity of the winning boat = (Total distance / Total time)
= (84 km / 1 hour)
= 84 km/h
Therefore, the answer to part (b) is that the average velocity of the winning boat is 84 km/h.