Questions: A swimming pool whose volume is 10,000 gal contains water that is 0.03% chlorine. Starting at t=0, city water containing 0.003% chlorine is pumped into the pool at a rate of 5 gal / min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hour? When will the pool water be 0006% chlorine? What is the percentage of chlorine in the pool after 1 hour? After 1 hour the pool will be % chlorine. (Round to four decimal places as needed)

Solution Steps

To solve this problem, we need to set up a differential equation that models the change in chlorine concentration over time. The rate of change of chlorine in the pool will depend on the rate at which chlorine is added and removed. We can then solve this differential equation to find the chlorine concentration after 1 hour.

1. Define the initial conditions and parameters.
2. Set up the differential equation for the chlorine concentration.
3. Solve the differential equation using Python.
4. Evaluate the solution at t = 1 hour.

We start with a swimming pool of volume \( V = 10,000 \, \text{gal} \) containing water with an initial chlorine concentration of \( C_{\text{initial}} = 0.03\% = 0.0003 \). City water with a chlorine concentration of \( C_{\text{city}} = 0.003\% = 0.00003 \) is pumped into the pool at a rate of \( 5 \, \text{gal/min} \).

The rate of change of chlorine concentration \( C(t) \) in the pool can be modeled by the differential equation: \[ \frac{dC}{dt} = \frac{Q}{V} (C_{\text{city}} - C) \] where \( Q = 5 \, \text{gal/min} \) is the flow rate and \( V = 10,000 \, \text{gal} \) is the volume of the pool.

We solve the differential equation over a time period of \( t = 60 \, \text{min} \). The solution gives us the chlorine concentration at \( t = 60 \, \text{min} \).

After solving the differential equation, we find that the chlorine concentration after 1 hour is: \[ C(60) \approx 0.0002920202943418588 \] Converting this to a percentage: \[ C(60) \times 100 \approx 0.0292\% \]

Final Answer