A tourist traveled on a motorboat against the current for 25 km. And then returned back on a raft. In the boat the tourist traveled for 10 hours less than on the raft. Find the speed of the current if the speed of the motorboat in still water is 12 km/hour.
I will assume that the speed of the raft is the speed of the current.
Let the speed of the current be x km/h
speed of boat against current = 12-x km/h
time with boat = 25/(12-x)
time on raft = 25/x
25/x - 25(12-x) = 10
divide by 5
5/x - 5/(12-x) = 2
multiply each term by x(12-x)
5(12-x) - 5x = 2x(12-x)
60 - 5x - 5x = 24x - 2x^2
2x^2 - 34x + 60 = 0
x^2 - 17x + 30 = 0
(x-2)(x-15) = 0
x = 2 or x = 15
but at a current of 15 km/h the boat would be going backwards, so
x = 2 , the speed of the current.
To solve this problem, let's break it down step by step:
Step 1: Let's assume the speed of the current is "c" km/hour.
Step 2: Since the speed of the motorboat in still water is given as 12 km/hour, the speed of the motorboat against the current would be (12 - c) km/hour.
Step 3: The time taken to travel 25 km against the current can be calculated using the formula: time = distance / speed. Therefore, the time taken on the motorboat would be 25 / (12 - c) hours.
Step 4: We are given that the time taken on the raft is 10 hours more than on the motorboat. So, the time taken on the raft would be (25 / (12 - c)) + 10 hours.
Step 5: The speed of the motorboat with the current can be calculated using the formula: speed of motorboat with current = speed of motorboat in still water + speed of current. Therefore, the speed of the motorboat with the current is (12 + c) km/hour.
Step 6: The time taken to return on the raft can be calculated using the formula: time = distance / speed. Therefore, the time taken on the raft would be 25 / (12 + c) hours.
Step 7: We are given that the time taken on the raft is (25 / (12 - c)) + 10 hours. Therefore, we can set up the equation:
25 / (12 + c) = (25 / (12 - c)) + 10
Step 8: Simplify the equation by multiplying both sides by (12 + c)(12 - c).
25(12 - c) = 25(12 + c) + 10(12 + c)(12 - c)
Step 9: Expand and simplify the equation:
300 - 25c = 300 + 25c + 120c - 10c^2
Step 10: Rearrange the equation to isolate the quadratic term on one side:
10c^2 + 145c - 300 = 0
Step 11: Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
Step 12: Once the quadratic equation is solved, we will have two possible values for "c," representing the speed of the current. One value may be positive, and the other may be negative.
Therefore, the speed of the current can be determined by solving the quadratic equation obtained in Step 11.