8. 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 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? *
Answer: the speed of the boat on the lake is 9 mph Step-by-step explanation: Let x represent the speed of the boat on the lake or in still water. The speed of the current in a river is 6 mph. This means that if the boat goes upstream against the speed of the current, its total speed would be (x - 6)mph. If the boat goes downstream against the speed of the current, its total speed would be (x + 6)mph. Time = distance/ speed 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. This means that the time taken to travel upstream is 22.5/(x - 6). The time taken to travel downstream is 22.5/(x + 6) Since the total time is 9 hours, it means that 22.5/(x - 6) = 22.5/(x + 6) Cross multiplying, it becomes 22.5(x + 6) + 22.5(x - 6) = 9 Multiplying through by (x + 6)(x - 6), it becomes 22.5(x - 6) + 22.5(x + 6) = 9[(x + 6)(x - 6)] 22.5x - 135 + 22.5x + 135 = 9(x² - 6x + 6x - 36) 22.5x + 22.5x = 9x² - 324 9x² - 45x - 324 = 0 Dividing through by 9, it becomes x² - 5x - 36 = 0 x² + 4x - 9x - 36 = 0 x(x + 4) - 9(x + 4) = 0 x - 9 = 0 or x + 4 = 0 x = 9 or x = - 4 Since the speed cannot be negative, then x = 9
