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 can use the concept of relative velocity. Let's analyze the situation step by step:
Step 1: Find the time taken by boat A to cross the lake and return.
The distance across the lake is 42 km. Since boat A travels at a constant speed of 42 km/h, the time taken to cross the lake is given by:
Time taken for crossing = Distance / Speed = 42 km / 42 km/h = 1 hour
Since the boat also returns at the same speed, the total time for the round trip is:
Total time for round trip = 1 hour + 1 hour = 2 hours
Step 2: Find the time taken by boat B to cross the lake and return.
The distance across the lake is still 42 km. Boat B travels with a speed of 21 km/h. To find the time taken to cross the lake, we can use the same formula as above:
Time taken for crossing = Distance / Speed = 42 km / 21 km/h = 2 hours
However, on the return journey, boat B increases its speed to 63 km/h. So, the time taken to return is given by:
Time taken for return = Distance / Speed = 42 km / 63 km/h = 0.67 hours
Total time for round trip = 2 hours + 0.67 hours = 2.67 hours
Step 3: Compare the total times taken by both boats.
We can see that boat A took 2 hours for the round trip, while boat B took 2.67 hours. Therefore, boat A wins the race.
Step 4: Calculate the difference in time between the two boats.
Difference in time = Total time taken by boat B - Total time taken by boat A
Difference in time = 2.67 hours - 2 hours = 0.67 hours
Step 5: Calculate the average velocity of the winning boat.
Average velocity = Total distance covered / Total time taken
Total distance covered = Distance across the lake + Distance back = 42 km + 42 km = 84 km
Average velocity = 84 km / 2 hours = 42 km/h
So, the answers to the given questions are:
(a) Boat A wins.
(b) Boat A wins by 0.67 hours.
(c) The average velocity of the winning boat is 42 km/h.
Please note that these calculations assume that both boats maintain a constant speed throughout their journeys.
To solve this problem, we need to calculate the total time taken by each boat to complete the round trip. Let's start with Boat A:
1. Boat A's speed across the lake is 42 km/h. So, the time it takes to cross the lake is given by distance/speed = 42/42 = 1 hour.
2. Boat A's speed on the return journey is also 42 km/h. Therefore, the time it takes to return is also 1 hour.
3. Total time taken by Boat A = Time to cross + Time to return = 1 + 1 = 2 hours.
Now let's calculate the total time taken by Boat B:
1. Boat B's speed across the lake is 21 km/h. So, the time it takes to cross the lake is given by distance/speed = 42/21 = 2 hours.
2. Boat B's speed on the return journey is 63 km/h. Therefore, the time it takes to return is given by distance/speed = 42/63 = 2/3 hours.
3. Total time taken by Boat B = Time to cross + Time to return = 2 + 2/3 = 8/3 hours.
Now we can compare the total times taken by both boats:
Total time taken by Boat A = 2 hours
Total time taken by Boat B = 8/3 hours ≈ 2.667 hours
Since Boat A takes less time to complete the round trip, Boat A wins.
(a) The winner is Boat A.
To calculate by how much Boat A wins, we need to find the difference in total times between the two boats:
Difference in time = Total time taken by Boat B - Total time taken by Boat A
= (8/3) - 2
= (8 - 6)/3
= 2/3 hours
(b) The average velocity of the winning boat can be calculated by dividing the total distance traveled (84 km = 42 km each way) by the total time taken (2 hours).
Average velocity of the winning boat = Total distance / Total time taken
= 84 km / 2 hours
= 42 km/h
So, the average velocity of the winning boat is 42 km/h.