Step 1 :Let's denote the speed of the boat in still water as 'b' and the speed of the river current as 'r'.
Step 2 :When the boat travels from Town A to Town B along the river's current, the effective speed of the boat is (b + r). Given that the distance is 198 miles and the time taken is 11 hours, we can write the equation as: \((b + r) * 11 = 198\).
Step 3 :When the boat travels back from Town B to Town A against the river's current, the effective speed of the boat is (b - r). Given that the distance is 198 miles and the time taken is 16.5 hours, we can write the equation as: \((b - r) * 16.5 = 198\).
Step 4 :Solving this system of equations, we find that 'b' equals 15 and 'r' equals 3.
Step 5 :Final Answer: The boat's speed in still water with the engine running at full speed is \(\boxed{15 \, \text{mi/hr}}\) and the river current's speed was \(\boxed{3 \, \text{mi/hr}}\).