b ,c please include the process of substitution to find expression of time for both

and also d

thankyou

In fluid dynamics, we use the term convection to refer to the transfer of fluid in a system due to its motion. We can use an ordinary differential equation to model the transport within the system of such fluids (e.g. water, gasoline) or substances dissolved in these fluids (e.g. salt, oxygen, pollutants), by modelling the change in the amount of fluid (or substance) with respect to time as the balance between inflowing and outflowing fluid. One application of such models is to study the environmental impact of large chemical spills, caused by industrial accidents, in order to manage them better. Suppose that due to an environmental accident, polluted water starts to flow into a lake of volume V [m] at a constant flowraten (mn). Water leaves this lake through a channel at a constant flowrate [mn). The concentration of pollutant in the incoming water is a [kg m). The change in the amount of pollutant y deposited in this lake as a function of time t is given by a differential equation of the type: dy (input rate of pollutant) - (output rate of pollutant) a) (15%) Derive a mathematical model for the amount of pollutant y(t) [kg] deposited in the lake as a function of time by solving the differential equation above. Use dimensional analysis to show your model is dimensionally correct. b) (15%) Knowing that this lake's ecosystem will be irreversibly damaged when the concentration of pollutant reaches 331 g m, derive an expression for the total time that local | authorities have to take action on this issue from the moment pollution starts flowing into this lake. Support your answer with a graph. c) [40%] Derive a mathematical expression for the time it takes until the concentration of pollutant in the lake, cu(t) = y), reaches 95% of its steady state concentration given by: yt) lim Indicate what the region of steady state concentration is in a plot of cu(t). By analysing the curve for cu(t), describe what is a steady state regime in your own words. Discuss how the lake's volume and the balance between n and r, affect the time that it takes for the concentration of pollutant in the lake to reach steady state. d) (30%) If local authorities act quickly enough and pollution into the lake is stopped when the pollutant concentration in the lake is at 213 g m, calculate how long it will take for the lake to clear out and have pollutant concentrations