Find. The solids retention time (SRT, $\theta_c$); if it is short of typical design practice, how it can be increased.
Approach. The return-sludge (RAS) concentration $X_r$ is not given directly, so recover it from a clarifier solids balance after first recovering the effluent flow $Q_e$ from an overall system flow balance; then apply the standard SRT definition.
Recover the effluent flow. At steady state the only flows crossing the SYSTEM boundary (reactor + clarifier together) are the influent in and the effluent plus waste out — the recycle flow $Q_r$ is entirely internal and does not appear in this balance: $$Q = Q_e+Q_w \;\Rightarrow\; Q_e = 10{,}000-300 = \boxed{9{,}700\ \text{gal/day}}$$
Recover the RAS/underflow concentration $X_r$ from a clarifier solids balance. Solids entering the clarifier (at flow $Q+Q_r$, concentration $X$) equal solids leaving as effluent ($Q_e,X_e$) plus underflow ($Q_r+Q_w$, concentration $X_r$): $$(Q+Q_r)X = Q_eX_e+(Q_r+Q_w)X_r$$ $$(10{,}000+5{,}000)(2000) = (9{,}700)(40)+(5{,}000+300)X_r$$ $$30{,}000{,}000 = 388{,}000+5{,}300\,X_r \;\Rightarrow\; X_r = \boxed{5{,}587\ \text{mg/L}}$$
Solids retention time. $\theta_c$ is the mass of solids held in the system (reactor) divided by the rate solids LEAVE the system (in the waste stream and the effluent, the only two exits crossing the system boundary): $$\theta_c = \frac{VX}{Q_wX_r+Q_eX_e} = \frac{(5{,}000)(2000)}{(300)(5{,}587)+(9{,}700)(40)} = \frac{10{,}000{,}000}{1{,}676{,}100+388{,}000} = \frac{10{,}000{,}000}{2{,}064{,}100} = \boxed{4.85\ \text{days}}$$
Assess and recommend. A conventional activated-sludge process is typically designed for an SRT of about 5–15 days for reliable BOD/TOC removal (and 10–20+ days if nitrification is required); at 4.85 days this system's SRT is at or slightly below the low end of that range, so it is effectively too short for anything beyond basic organic-carbon removal. Since $\theta_c=VX/(Q_wX_r+Q_eX_e)$ is dominated by the $Q_wX_r$ term, the most direct and immediately controllable fix is to reduce the wasting rate $Q_w$ (waste sludge less frequently/at a lower rate), which lengthens $\theta_c$ almost in direct inverse proportion; increasing the reactor volume $V$ or operating at a higher MLVSS $X$ (by returning more solids, i.e. raising $Q_r$/$X_r$) would also increase $\theta_c$, but reducing $Q_w$ is the standard day-to-day operational lever.
