Officials begin to release water from a full man-made lake at a rate that would empty the lake in 18 weeks, but a river that can fill the lake in 25 weeks is replenishing the lake at the same time. How many weeks does it take to empty the lake? Express your answer as a fraction reduced to lowest terms, if needed.
Solution
Step 1 ：The rate at which the lake is being emptied is \(\frac{1}{18}\) per week, and the rate at which the lake is being filled is \(\frac{1}{25}\) per week.
Step 2 ：The net rate at which the lake is being emptied is the difference between these two rates. We can find this rate by subtracting the filling rate from the emptying rate.
Step 3 ：Once we have the net rate, we can find the time it takes to empty the lake by taking the reciprocal of this rate.
Step 4 ：Calculating the emptying rate gives us approximately 0.05555555555555555.
Step 5 ：Calculating the filling rate gives us approximately 0.04.
Step 6 ：Subtracting the filling rate from the emptying rate gives us a net rate of approximately 0.015555555555555552.
Step 7 ：Taking the reciprocal of the net rate gives us the time it takes to empty the lake, which is approximately 64.2857142857143 weeks.
Step 8 ：Final Answer: It takes approximately \(\boxed{64.2857142857143}\) weeks to empty the lake.
