A truck carrying $2500$ kg container of a chemical crashes into a reservoir. The container ruptures in a catastrophic manner instantaneously. The chemical instantly forms a layer at the bottom of the lake $(100$ m$^2).$ The diffusion coefficient of the chemical is $10-6$ m$2/$s and its solubility is $50$ ppm$.$ The lake is $10$ meters deep and none of the chemical will volatilize from its surface. A$)$ Please determine the $1-$D concentration profile in this lake at t $=10$ days using both the analytical solution and the numerical solution provided in$-$class $($With a boundary and without$).$ Please compare the results to determine which, if any is more accurate. B$)$ Please determine the length of time required for the water column to be completely mixed $(<5\%$ difference between top and bottom concentrations$).$ C$)$ Please play around with the diffusion coefficients and the values of dt and discuss what happens to the prediction and why you think it happens.