$($Problem $4\mathrm{.}7$ in Surface Water$-$Quality Modeling$)$ A well$-$mixed lake with the following

characteristics has a steady$-$state concentration of $5\frac{g}{L}$ of total phosphorus.

Inflow $=$ Outflow $=5{10}^5\frac{m^3}{y}r$

Volume $=4{10}^7{m}^3$

Surface area $=5{10}^6{m}^2$

At the beginning of $1994$ it received an additional phosphorus loading of $500k\frac{g}{y}r$ from a

fertilizer processing plant. Then in $1997,$ a small subdivision was located on the lake.

Population of this subdivision can be described using the following exponential model:

$p=200e^{G_{p}t}$

where $G_p=$ first$-$order population growth rate $=\frac{0\mathrm{.}2}{y}r.$ If total phosphorus settles at a rate of $8$

$\frac{m}{y}r$ and assuming each person generates $0\mathrm{.}5k\frac{g}{y}r$ of total phosphorus, determine the resultant

phosphorus concentration in the lake from $1994$ to $2010.$