$(15$ points, Estimated time $20$min $)$ Just as you are leaving town for a weekend ski trip, you discover a slow leak in your goldfish aquarium. You make your best estimate $($by eye$)$ of the leak flowrate, $Q_{\text{o}\text{u}\text{t}\text{.}}.$ You then move the tank under your sink faucet and set the faucet to a slow drip $(Q_{in})$ so to maintain the aquarium water level at its current level. Unfortunately, you made an underestimate, and the tank level begins dropping toward the minimum level needed to keep your fish alive $($i$.$e$.,$ the danger level, $z_d).$

Starting with the Reynolds transport theorem, derive an expression for how the aquarium height changes with time. Use the variables names below and the variable $z$ for the water level.

Using the values below, determine the amount of time $($in hours$)$ needed for the tank to drain to the danger level. Report your time to the nearest $0\mathrm{.}1h.$

Assume

The aquarium has a square base, with a width $w=25cm$

$Q_{\text{o}\text{u}\text{t}}=15c\frac{m^3}{m}in$

$Q_{in}=14c\frac{m^3}{m}in$

The tank level is currently

$z_c=37cm$

The danger level is $z_d=3cm$