Question

How do you determine what percentage of chlorine is in the pool after $1$ hour if a pool whose volume is $10000$ gallons contains water the is $0.01\% $ chlorine and starting at $t = 0$ city water containing $0.001\% $ chlorine is pumped into the pool at a rate of $5\,gal\;{\min ^{ - 1}}$ while the pool water flows out at the same rate?

Answer 1

Hint: To solve this question we have to assume the amount of chlorine in the pool at the time $t$ be $x$. By using the rate of change of chlorine in the pool, we will find the amount of the chlorine. After that we find the required answer using it.

 Complete answer:
Let us take $x$ the amount of chlorine that is in the pool at time $t$.
Then the rate of change of $x$ is:
$\dfrac{{dx}}{{dt}} = \left( {rate\;in} \right) - \left( {rate\;out} \right)$
$\dfrac{{dx}}{{dt}} = {c_i} \times {f_{in}} - \dfrac{{x\left( t \right)}}{{10000}} \times {f_{out}}$
${c_i}$ is the concentration of chlorine of the inflow, $0.001\% $. ${f_{in}}$ is the inflow, $5\;gal\,{\min ^{ - 1}}$and ${f_{out}}$ is the outflow, $5\;gal\;{\min ^{ - 1}}$.
$\dfrac{{dx}}{{dt}} = \dfrac{{0.001}}{{100}} \times 5 - \dfrac{x}{{10000}} \times 5$
$ = \dfrac{1}{{20000}} - \dfrac{x}{{2000}}$
$ \Rightarrow - \dfrac{{x - 0.1}}{{2000}}$
$ \Rightarrow \dfrac{{dx}}{{x - 0.1}} = \dfrac{{ - dt}}{{2000}}$
$ \Rightarrow \ln \left| {x - 0.1} \right| = \dfrac{{ - t}}{{2000}} + c$
$ \Rightarrow x - 0.1 = c \times {e^{ - \dfrac{t}{{2000}}}}$
n $t = 0$, the concentration is $0.01\% $, so amount of chlorine $x$ is
$\dfrac{{0.01}}{{100}} \times 10000 = 1\;gal$
Accordingly in the equation, we have
$x = 0.1 + C \times {e^{\dfrac{{ - t}}{{2000}}}}$
$1 = 0.1 + c \times {e^{ - \dfrac{0}{{2000}}}}$
$ \Rightarrow 1 = 0.1 + c \times 1$
$ \Rightarrow c = 0.9$
The amount of chlorine for any time $t$ is then
$x\left( t \right) = 0.1 + 0.9 \times {e^{ - \dfrac{t}{{2000}}}}$
At one hour $\left( {t = 60\;\min } \right)$, the amount of chlorine is,
$x\left( {60} \right) = 0.1 + 0.9 \times {e^{ - \;\dfrac{{60}}{{2000}}}}$
$ = 0.1 + 0.9 \times 0.9704$
$ = 0.973$
This amount means a concentration of
$c = \dfrac{A}{{10000}} = \dfrac{{0.973}}{{10000}}$
$ = 0.00973\% $
This is the required answer.

 Note:
Remember to check if there is no error in the calculation. Also look at the value of each and every term whether it is correct or wrong. It is also important to check whether the formula we are using is correct or wrong. After checking each of the above-mentioned problems we can get the right answer.