2 hour 40 min after a raft left pier A and traveled downstream, a motorboat left pier B and traveled upstream toward the raft. The two met 27 km away from B. Find the speed of the raft if the speed of the motorboat in still water is 12 km/hour and the distance from A to B is 44 km.
If the boat went 27 km, the raft went 17 km when they met.
let the speed of the current (as well as the raft) be x km/h
then the speed of the boat going upstream is 12-x km/h
time taken by the raft = 17/x
time taken by the boat = 27/(12-x)
The difference in their times = 2 2/3 hrs = 8/3 hrs
17/x - 27/(12-x) = 8/3
times 3x(12-x)
51(12-x) - 81x = 8x(12-x)
612 - 51x - 81x = 96x - 8x^2
8x^2 - 228x + 612 = 0
divide by 4
2x^2 - 57x + 153 = 0
(x-3)(2x - 51) = 0
x = 3 or x = 51/2, but x < 12 or else the boat would be going backwards.
the speed of the raft or the speed of the current is 3 km/h
Thanks for helping me in RSM
This answer is right!
wow good job with that explanation
To find the speed of the raft, we can make use of the relative speed concept.
Let's start by breaking down the information given:
- The motorboat's speed in still water is 12 km/hour.
- The distance between Pier A and Pier B is 44 km.
- The raft and the motorboat meet 27 km away from Pier B.
Now, let's denote the speed of the raft as 'r' km/hour.
When the motorboat is traveling upstream, it moves against the current. The speed of the current is the difference between the speed of the motorboat in still water and the speed of the raft.
So, the speed of the motorboat upstream is (12 - r) km/hour.
Similarly, when the raft is traveling downstream, it moves with the current. The speed of the current is the difference between the speed of the raft and the speed of the motorboat in still water.
So, the speed of the raft downstream is (r - 12) km/hour.
Since they meet 27 km away from Pier B, we can use the formula:
Time = Distance / Speed
To find the time it takes for both the raft and the motorboat to meet, we can set up two equations.
The time taken by the motorboat traveling upstream:
Time upstream = 27 km / (12 - r) km/hour
The time taken by the raft traveling downstream:
Time downstream = 44 km / (r - 12) km/hour
The total time taken is the sum of the time upstream and the time downstream:
Total time = Time upstream + Time downstream
The total time is given as 2 hours and 40 minutes, which is equivalent to 2.67 hours:
2.67 hours = Time upstream + Time downstream
Now, we can substitute the equations for the time upstream and time downstream:
2.67 hours = 27 km / (12 - r) km/hour + 44 km / (r - 12) km/hour
To solve this equation, we can clear the denominators by multiplying through by the denominators:
2.67 hours * (12 - r) * (r - 12) = 27 km * (r - 12) + 44 km * (12 - r)
Now, simplify and rearrange the equation:
2.67(144 - 12r - 12r + r^2) = 27(r - 12) + 44(12 - r)
2.67(144 - 24r + r^2) = 27r - 324 + 528 - 44r
Now, expand and collect like terms:
385.68 - 64.08r + 2.67r^2 = -17r + 204
Rearrange the equation to become a quadratic equation:
2.67r^2 - 46.92r + 181.68 = 0
At this point, we have a quadratic equation, and we can solve it using the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2.67, b = -46.92, and c = 181.68.
Let's substitute these values into the quadratic formula:
r = (-(-46.92) ± √((-46.92)^2 - 4 * 2.67 * 181.68)) / (2 * 2.67)
Solve this equation using a calculator or any numerical method, we find:
r ≈ 7.47 km/hour (approximately)
So, the speed of the raft is approximately 7.47 km/hour.