Hydraulic Reservoir Surge Control (Civil Engineering)

In the Las Vegas Valley, the Clark County Regional Flood Control District manages regional detention basins designed to capture severe monsoon flash floods. A concrete-lined flood detention channel is modeled geometrically as a horizontal symmetric right triangular prism.

The cross-section is an inverted isosceles triangle where the top surface width $w$ is always equal to the water depth $h$ ($w = h$). The total longitudinal length of the basin is a fixed $L = 100.0\text{ meters}$. The maximum design spillway height is $H_{\text{spillway}} = 6.0\text{ meters}$.

During an intense desert cloudburst storm event, stormwater runoff flows into the basin at a measured volumetric inflow rate of:

1. Derive the formula for the volume $V(h)$ of stormwater trapped in the basin as a function of water depth $h$, utilizing the geometric cross-sectional constraint $w = h$.
2. Apply implicit differentiation with respect to time $t$ to determine the equation relating the volumetric inflow rate $\frac{dV}{dt}$ and the rate of water level rise $\frac{dh}{dt}$.
3. Calculate the exact vertical rate at which the water level is rising ($\frac{dh}{dt}$) in $\text{m/min}$ and $\text{cm/min}$ at the instant the water depth reaches $h = 4.0\text{ meters}$.
4. If hydraulic safety protocols mandate activating downstream emergency bypass gates whenever the estimated time to reach the spillway crest ($h = 6.0\text{ m}$) falls below 25 minutes, determine whether civil hydrologists must open the emergency gates at the $h = 4.0\text{ m}$ checkpoint.