The speed of the current in a river is 6 mph. A ferry operator who works that part of the river is looking to buy a new boat for his business. Every day, his route takes him 22.5 miles each way against the current and back to his dock, and he needs to make this trip in a total of 9 hours. He has a boat in mind, but he can only test it on a lake where there is no current. How fast must the boat go on the lake in order for it to serve the ferry operator’s needs?
Let the speed of the boat in still water be x mph.
When the ferry is going against the current, the effective speed is (x - 6) mph.
Therefore, the total time taken to travel 22.5 miles against the current is:
22.5/(x-6)
When the ferry is going with the current, the effective speed is (x + 6) mph.
Therefore, the total time taken to travel 22.5 miles with the current is:
22.5/(x+6)
The total time taken for the round trip is given to be 9 hours. Therefore, we have:
22.5/(x-6) + 22.5/(x+6) = 9
Solving this equation gives: x = 24 mph
Therefore, the boat must go 24 mph on the lake in order to serve the ferry operator's needs.