Suppose that due to an environmental accident, polluted water starts to flow into a lake of volume V [m³] at a constant flowrate r [m³ h¹]. Water leaves this lake through a channel at a constant flowrate r, [m³ h¹]. The concentration of pollutant in the incoming water is c₂ [kg m³]. 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: b) c) a) dy dt (input rate of pollutant) - (output rate of pollutant) Derive a mathematical model for the amount of pollutant y(t) [kg] deposited in the lake as a function of time. You will need to justify your model in terms of dimensional analysis and solution methods for ordinary differential equations. Knowing that this lake's ecosystem will be irreversibly damaged when the concentration of pollutant reaches 331 g m³, calculate 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. Derive a mathematical expression for the time it takes until the concentration of pollutant in the lake, c₁ (t) = x(t), reaches 95% of its steady state concentration given by: y(t) lim t-00 V Indicate what the region of steady state concentration is in a plot of c(t). By analysing the curve for c₁(t), describe what is a steady state regime in your own words. Discuss how the lake's volume and the balance between r, and r, affect the time that it takes for the concentration of pollutant in the lake to reach steady state. If local authorities act quickly enough and pollution into the lake is stopped when the pollutant concentration in the lake is at 213 g m³, calculate how long it will take for the lake to clear out and have pollutant concentrations < 5 g m³. Design a mathematical model for the clearing of the lake and support your answer with a graph.