Question

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?The chapter I am working on is solving rational equations. Can you help me understand this question better?

Answer 1

Solution:

Given:

Round trip distance to and fro = 22.5 miles.

Total round trip time to and fro = 9 hours;

Thus; Let speed in still water = u mph

Upstream speed ( against the current ) = u - 6 mph

Downstream speed ( with the current ) = u + 6 mph

Thus, the time equation is;

$\begin{gathered} speed=\frac{distance}{time} \\ time=\frac{distance}{speed} \end{gathered}$

$\begin{gathered} \frac{22.5}{u-6}+\frac{22.5}{u+6}=9................multiply\text{ all through by }2(u-6)(u+6) \\ 45(u+6)+45(u-6)=18(u+6)(u-6) \\ 90u=18u^2-648 \\ 18u^2-90u+648=0 \end{gathered}$

Solving this quadratic, we have;

$u=9\text{ }or\text{ }u=-4$

Thus, the speed is 9mph.

Thus if the boat is to serve the ferry operator’s needs, it should move at 9mph