Problem $3.(12$ pts$)$ The ultimate BOD of a river just below a sewage outfall is $50m\frac{g}{L}$ and the DO is at the saturation value of $10m\frac{g}{L}.$ The deoxygenation rate coefficient is $0\mathrm{.}3da{y}^{-1}$ and the reaeration rate coefficient, $k_r,$ is $0\mathrm{.}9$ day ${?}^{-1}.$ The river is flowing at $48m\frac{i}{d}.$ Assume the only source of BOD is this outfall and the initial oxygen deficit $(D_0)$ is zero.

$($a$)$ Find the critical distance $($in miles$)$ downstream at which DO is a minimum.

$($b$)$ Find the minimum DO$.$

$($c$)$ If a wastewater treatment plant is to be built to treat the sewage, what fraction of the BOD would have to be removed to assume a minimum of $5m\frac{g}{L}$ downstream? Hint: calculate the $L_0$ under this design constraint $(D{O}_{\text{m}\text{i}\text{n}}=5m\frac{g}{L}).$ Then calculate the $\frac{L_{0,\text{n}\text{e}\text{w}}}{L_{0,\text{o}\text{l}\text{d}}}$ to determine how much needs to be removed $(1-\frac{L_{0,\text{n}\text{e}\text{w}}}{L_{0,\text{o}\text{l}\text{d}}}).$