A group of campers and one group leader left a campsite in a canoe. They traveled at an average rate of10km/h. Two hours later, the other group leader left the campsite ina motorboat. He traveled at an average rate of 22km/h.
a. How long after the canoe left the campsite did the motorboat catch up with it?
b. How long did the motorboat travel?
a 2 hr head start for the canoe is a 20 km lead
the boat closes the gap at 12 kph (22 - 10)
a. 2 hr + (20 km / 12 kph)
b. 20 km / 12 kph
Speed:
v = s / t
s = distance
t = time
t = s / v
t1 = travel time of group with canoe
t2 = travel time of group with motorboat
t1 = s / v1
t1 = s / 10
t2 = 2 hours + s / 22
t2 = 2 + s / 22
The motorboat catch up canoe when:
t1 = t2
s / 10 = 2 + s / 22
Subtract s / 22 to both sides
s / 10 - s / 22 = 2
11 ∙ s / 11 ∙ 10 - 5 ∙ s / 5 ∙ 22 = 2
11 s / 110 - 5 s / 110 = 2
6 s / 110 = 2
Multiply both sides by 110
6 s = 220
Divide both sides by 6
s = 220 / 6 = 2 ∙ 100 / 2 ∙ 3 = 110 / 3 km = 36.666 km
t = t1 = s / v1
t = ( 110 / 3 ) / 10 = 110 / 30 = 10 ∙ 11 / 10 ∙ 3 = 11 / 3 = 9 / 3 + 2 / 3 = 3 + 2 / 3
3 h + 2 h / 3 = 3 h + 2 ∙ 60 min / 3 = 3 h + 2 ∙ 20 min = 3 h 40 min
OR
t = t2 = 2 + s / 22 = 2 + ( 110 / 3 ) / 22 = 2 + 110 / 66 =
2 + 22 ∙ 5 / 22 ∙ 3 = 2 + 5 / 3 = 2 + 3 / 3 + 2 / 3 = 2 + 1 + 2 / 3 = 3 + 2 /3
= 3 + 2 ∙ 60 min / 3 = 3 h + 2 ∙ 20 min = 3 h 40 min
a.
3 h 40 min
b.
110 / 3 km = 36.666 km
To solve this problem, we need to use the concept of relative motion. Let's break it down step by step:
a. How long after the canoe left the campsite did the motorboat catch up with it?
Step 1: Find the head start distance (the distance the canoe traveled before the motorboat started).
The head start distance can be calculated by multiplying the average speed of the canoe (10 km/h) by the time it traveled before the motorboat started (2 hours).
Head start distance = Speed × Time
Head start distance = 10 km/h × 2 hours = 20 km
Step 2: Find the relative speed between the motorboat and the canoe.
The relative speed is the difference in speed between the motorboat (22 km/h) and the canoe (10 km/h).
Relative speed = Speed of motorboat - Speed of canoe
Relative speed = 22 km/h - 10 km/h = 12 km/h
Step 3: Calculate the time it takes for the motorboat to catch up with the canoe.
The time it takes for the motorboat to catch up with the canoe can be found by dividing the head start distance (20 km) by the relative speed (12 km/h).
Time = Distance / Speed
Time = 20 km / 12 km/h ≈ 1.67 hours
So, the motorboat caught up with the canoe approximately 1.67 hours after it left the campsite.
b. How long did the motorboat travel?
To find out how long the motorboat traveled, we need to add the time it took for the motorboat to catch up with the canoe to the 2-hour head start time.
Total travel time of the motorboat = Time taken to catch up + Head start time
Total travel time of the motorboat = 1.67 hours + 2 hours = 3.67 hours
Therefore, the motorboat traveled for approximately 3.67 hours.