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 this please help!
I am very confused with this problem and as to how the answer is found but the correct answer is the following
boat a wins
by 42km
and the average velocity of the winning boat is 0(zero). I got it right by practicing a webassign problem.
To solve this problem, we will calculate the time it takes for each boat to complete the round-trip and compare the results.
Boat A:
- Going across the lake: distance = 42 km, speed = 42 km/h
- Time taken = distance / speed = 42 km / 42 km/h = 1 hour
- Returning back: distance = 42 km, speed = 42 km/h
- Time taken = distance / speed = 42 km / 42 km/h = 1 hour
- Total time taken = 1 hour + 1 hour = 2 hours
Boat B:
- Going across the lake: distance = 42 km, speed = 21 km/h
- Time taken = distance / speed = 42 km / 21 km/h = 2 hours
- Returning back: distance = 42 km, speed = 63 km/h
- Time taken = distance / speed = 42 km / 63 km/h = 2/3 hour
- Total time taken = 2 hours + 2/3 hour = 8/3 hours
Comparing the total time taken, we can see that Boat B completes the round-trip faster than Boat A. Therefore, Boat B wins the race.
To find the time difference:
- Boat A's total time taken = 2 hours
- Boat B's total time taken = 8/3 hours
- Difference = Boat B's total time taken - Boat A's total time taken
- Difference = (8/3) hours - 2 hours
To compute this, we need a common denominator:
- (8/3) hours = (8 * 60 / 3) minutes = 160/3 minutes
- 2 hours = 2 * 60 minutes = 120 minutes
Difference = (160/3 minutes) - 120 minutes
Difference = (160 - 360)/3 minutes
Difference = (-200)/3 minutes ≈ -66.67 minutes
Therefore, Boat B finishes the round-trip approximately 66.67 minutes (or 1 hour and 6.67 minutes) ahead of Boat A.
To find the average velocity of the winning boat (Boat B):
- Total distance covered = 42 km + 42 km = 84 km
- Total time taken = 8/3 hours
- Average velocity = total distance covered / total time taken
- Average velocity = 84 km / (8/3 hours) ≈ 31.5 km/h
Hence, Boat B wins the race by approximately 66.67 minutes and has an average velocity of approximately 31.5 km/h.
To determine which boat wins the race, we need to calculate the time it takes for each boat to complete the round trip.
Let's start by calculating the time it takes for Boat A to complete the round trip. Boat A travels at a speed of 42 km/h for both legs of the trip, so the time taken to cross the lake (one way) is given by the formula:
Time = Distance / Speed
Time taken for Boat A to cross the lake = 42 km / 42 km/h = 1 hour
Since the turnaround time is negligible, the total time taken for Boat A to complete the round trip is twice the time taken for crossing the lake:
Total time taken for Boat A = 2 hours
Now let's calculate the time it takes for Boat B to complete the round trip. Boat B travels at a speed of 21 km/h when crossing the lake, and then speeds up to 63 km/h during the return journey.
Time taken for Boat B to cross the lake = 42 km / 21 km/h = 2 hours
Time taken for Boat B to return = 42 km / 63 km/h = 0.67 hours
Total time taken for Boat B to complete the round trip = Time to cross + Time to return = 2 hours + 0.67 hours = 2.67 hours
Comparing the total times, we can see that Boat A takes 2 hours to complete the round trip, while Boat B takes 2.67 hours. Therefore, Boat A wins the race.
To answer the question "by how much," we need to calculate the time difference between the two boats. In this case, it is the difference in the total times taken:
Time difference = Total time taken for Boat B - Total time taken for Boat A = 2.67 hours - 2 hours = 0.67 hours
So, Boat A wins the race by 0.67 hours.
The average velocity of the winning boat, Boat A, can be calculated by dividing the total distance covered (round trip) by the total time taken. In this case, Boat A covers a distance of 42 km in a total time of 2 hours:
Average velocity of Boat A = Total distance / Total time = 42 km / 2 hours = 21 km/h
Therefore, the average velocity of the winning boat, Boat A, is 21 km/h.
I hope this helps! Let me know if you have any further questions.