answer and show all the steps that you use to solve this problem in the space provided.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's denote the speed of the boat in still water as B mph.
When the boat travels against the current, its effective speed is B-6 mph.
When the boat travels with the current, its effective speed is B+6 mph.
Given that the total distance of the trip is 22.5 miles each way, the total distance is 45 miles.
We are also given that the total time for the trip is 9 hours.
Using the formula distance = rate x time, we can set up the following equations:
22.5/(B-6) + 22.5/(B+6) = 9
To solve this equation, we multiply through by the common denominator (B-6)(B+6) to get:
22.5(B+6) + 22.5(B-6) = 9(B-6)(B+6)
Simplify the equation:
22.5B + 135 + 22.5B - 135 = 9(B^2 - 36)
Combine like terms:
45B = 9B^2 - 324
Rearrange to set the equation equal to zero:
9B^2 - 45B - 324 = 0
Divide by 9 to simplify:
B^2 - 5B - 36 = 0
This is a quadratic equation that can be factored:
(B - 9)(B + 4) = 0
Therefore, B = 9 or B = -4. Since the speed of the boat cannot be negative, the speed of the new boat on the lake must be 9 mph in order to meet the ferry operator's needs.