There is a perfectly mixed tank that contains initially $100m^3$ of sugar solution with a sugar concentration of $0\mathrm{.}8k\frac{g}{m^3}.$ This tank is heated from the bottom with the help of an electric heater and the steam is generated with a volumetric flow rate of $1\frac{m^3}{h}.$

a$)(10$p$)$ Derive the expression for the variation of the change in the volume of the tank with time

b$)(15$p$)$ Derive the expression for the variation of the change in the sugar concentration in the tank with time.

c$)(5$p$)$ Calculate the concentration of the mixture in the tank after $12$ hours

Data: ${}_{\text{s}\text{u}\text{g}\text{a}\text{r}\text{s}\text{o}\text{l}\text{u}\text{t}\text{i}\text{o}\text{n}}=1200k\frac{g}{m^3},$ water $=1000k\frac{g}{m^3}$