Answer:
![c(t) = 0.044\cdot e ^{-0.066\cdot t}](https://tex.z-dn.net/?f=c%28t%29%20%3D%200.044%5Ccdot%20e%20%5E%7B-0.066%5Ccdot%20t%7D)
Step-by-step explanation:
The quantity of salt inside the tank is modelled after the Principle of Mass Conservation:
Salt
![\dot m_{in, salt} - \dot m_{out, salt} = \frac{dm_{tank,salt}}{dt}](https://tex.z-dn.net/?f=%5Cdot%20m_%7Bin%2C%20salt%7D%20-%20%5Cdot%20m_%7Bout%2C%20salt%7D%20%3D%20%5Cfrac%7Bdm_%7Btank%2Csalt%7D%7D%7Bdt%7D)
Water
![\dot m_{in,water} - \dot m_{out,water} = \frac{dm_{tank,water}}{dt}](https://tex.z-dn.net/?f=%5Cdot%20m_%7Bin%2Cwater%7D%20-%20%5Cdot%20m_%7Bout%2Cwater%7D%20%3D%20%5Cfrac%7Bdm_%7Btank%2Cwater%7D%7D%7Bdt%7D)
Given that water is an incompressible fluid, the expression can be simplified into the following expression:
![\dot V_{in, water} - \dot V_{out,water} = \frac{dV_{tank, water}}{dt}](https://tex.z-dn.net/?f=%5Cdot%20V_%7Bin%2C%20water%7D%20-%20%5Cdot%20V_%7Bout%2Cwater%7D%20%3D%20%5Cfrac%7BdV_%7Btank%2C%20water%7D%7D%7Bdt%7D)
Both flows have the same rate and tank can be modelled as a steady state system.
![\dot V_{in, water} - \dot V_{out,water} = 0](https://tex.z-dn.net/?f=%5Cdot%20V_%7Bin%2C%20water%7D%20-%20%5Cdot%20V_%7Bout%2Cwater%7D%20%3D%200)
The expression for salt concentration in the tank is:
-![-\dot V_{tank}\cdot c = V_{tank} \cdot \frac{dc}{dt}](https://tex.z-dn.net/?f=-%5Cdot%20V_%7Btank%7D%5Ccdot%20c%20%3D%20V_%7Btank%7D%20%5Ccdot%20%5Cfrac%7Bdc%7D%7Bdt%7D)
After some handling, the following homogeneous first-order linear differential equation is found:
![\frac{V_{tank}}{\dot V_{tank}} \cdot \frac{dc}{dt} + c = 0](https://tex.z-dn.net/?f=%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%20%5Ccdot%20%5Cfrac%7Bdc%7D%7Bdt%7D%20%2B%20c%20%3D%200)
Where
. The solution is obtained by using Laplace transforms:
![\frac{V_{tank}}{\dot V_{tank}} \cdot \left[s\cdot C(s) - c(0)\right] + C(s) = 0](https://tex.z-dn.net/?f=%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%20%5Ccdot%20%5Cleft%5Bs%5Ccdot%20C%28s%29%20-%20c%280%29%5Cright%5D%20%2B%20C%28s%29%20%3D%200)
![\left(\frac{V_{tank}}{\dot V_{tank}}\cdot s + 1\right)\cdot C(s) = \frac{V_{tank}}{\dot V_{tank}}\cdot c(0)](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%5Ccdot%20s%20%2B%201%5Cright%29%5Ccdot%20C%28s%29%20%3D%20%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%5Ccdot%20c%280%29)
![C(s) = \frac{\frac{V_{tank}}{\dot V_{tank}}\cdot c(0) }{\left(\frac{V_{tank}}{\dot V_{tank}} \right)\cdot \left(s + \frac{\dot V_{tank}}{V_{tank}} \right)}](https://tex.z-dn.net/?f=C%28s%29%20%3D%20%5Cfrac%7B%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%5Ccdot%20c%280%29%20%7D%7B%5Cleft%28%5Cfrac%7BV_%7Btank%7D%7D%7B%5Cdot%20V_%7Btank%7D%7D%20%5Cright%29%5Ccdot%20%5Cleft%28s%20%2B%20%5Cfrac%7B%5Cdot%20V_%7Btank%7D%7D%7BV_%7Btank%7D%7D%20%20%5Cright%29%7D)
![c(t) = c(0) \cdot e^{-\frac{\dot V_{tank}}{V_{tank}}\cdot t }](https://tex.z-dn.net/?f=c%28t%29%20%3D%20c%280%29%20%5Ccdot%20e%5E%7B-%5Cfrac%7B%5Cdot%20V_%7Btank%7D%7D%7BV_%7Btank%7D%7D%5Ccdot%20t%20%7D)
Where
.
The formula for the concentration of salt in the tank is:
![c(t) = 0.044\cdot e ^{-0.066\cdot t}](https://tex.z-dn.net/?f=c%28t%29%20%3D%200.044%5Ccdot%20e%20%5E%7B-0.066%5Ccdot%20t%7D)