Hydroelectric Reservoir Net Change (Civil & Environmental Engineering) Solution

Tracking net water volume changes in hydroelectric reservoirs requires integrating the net flow rate (inflow minus spillway discharge) over an operational time period.

Water discharges through a spillway of a reservoir at a time-varying rate of $r(t) = 18 - \frac{t^2}{2}$ cubic meters per minute for $t \ge 0$, where $t$ is measured in minutes. Initially, the reservoir has $65$ cubic meters of active storage.

Determine the following:

1. Write the definite integral representing the total volume of water discharged (lost) from the reservoir between $t = 0$ and $t = 3$ minutes.
2. Evaluate this integral and determine how much water is remaining in the reservoir after $3$ minutes.

Step-by-Step Solution:

1. Write the Definite Integral: The rate of water leaving the reservoir is given by $r(t) = 18 - \frac{t^2}{2}$. The total volume of water discharged over the interval $[0, 3]$ is the definite integral:

   $$\text{Volume Discharged} = \int_{0}^{3} \left(18 - \frac{t^2}{2}\right) dt$$

2. Evaluate the Integral and Calculate Remaining Water: First, find the antiderivative of the rate function:

   $$\int \left(18 - \frac{t^2}{2}\right) dt = 18t - \frac{t^3}{6}$$

   Evaluate this antiderivative from $t = 0$ to $t = 3$:

   $$\left[18(3) - \frac{3^3}{6}\right] - \left[18(0) - \frac{0^3}{6}\right]$$
   $$= \left[54 - \frac{27}{6}\right] - 0$$
   $$= 54 - 4.5 = 49.5\text{ cubic meters}$$

   Subtract the discharged volume from the initial $65\text{ cubic meters}$ to find the remaining volume:

   $$\text{Volume Remaining} = 65 - 49.5 = 15.5\text{ cubic meters}$$

Meaning of the Answer & Real-Life Application:

Applying the Net Change Theorem allows engineers to monitor reservoir reserves and predict power generation capacities. Integrating flow rates ensures that water levels remain above minimum hydraulic intake heights to prevent turbine cavitation.