A tugboat can go 10 miles upstream in the same time as it can go 15 miles downstream. If the tugboat goes 12 miles per hour in still water, what is the speed of the current?
To find the speed of the current, we can set up equations to represent the given information. Let's denote the speed of the current as "c."
When the tugboat is moving upstream, it has to work against the current, so its effective speed would be the difference between the boat's speed in still water (12 mph) and the speed of the current. Therefore, the speed of the tugboat when moving upstream would be (12 - c).
Similarly, when the tugboat is moving downstream, it benefits from the current, so its effective speed would be the sum of the boat's speed in still water (12 mph) and the speed of the current. Therefore, the speed of the tugboat when moving downstream would be (12 + c).
Now, we know that the tugboat takes the same amount of time to travel 10 miles upstream and 15 miles downstream. The time it takes to travel a certain distance is given by the formula:
Time = Distance / Speed
Using this formula, we can set up the following equation for the time it takes the tugboat to travel upstream:
10 / (12 - c) = 15 / (12 + c)
Now, we can solve this equation to find the speed of the current (c). Let's proceed with solving it:
Cross-multiplying the equation gives us:
10(12 + c) = 15(12 - c)
Expanding both sides of the equation:
120 + 10c = 180 - 15c
Combining like terms:
10c + 15c = 180 - 120
25c = 60
Dividing both sides by 25:
c = 60 / 25
Simplifying the fraction gives us the final answer:
c = 2.4 mph
Therefore, the speed of the current is 2.4 miles per hour.