Problem 1: Muskingum Routing Coefficients Calculation A river reach has a storage constant ($K$) of 12 hours and a weighting factor ($x$) of 0.2. Calculate the Muskingum routing coefficients ($C_0$, $C_1$, and $C_2$) for a routing time step ($\Delta t$) of 6 hours. Verify that their sum equals 1.0.

Problem 2: Muskingum Channel Routing Application Given the routing coefficients calculated in Problem 1 ($C_0 = 0.048$, $C_1 = 0.429$, $C_2 = 0.524$), calculate the outflow ($O_2$) at $t=6 \text{ hours}$. The initial conditions at $t=0$ are: Inflow ($I_1$) = $20 \text{ m}^3/\text{s}$ and Outflow ($O_1$) = $20 \text{ m}^3/\text{s}$. The measured inflow at $t=6 \text{ hours}$ ($I_2$) is $50 \text{ m}^3/\text{s}$.

Problem 3: Reservoir Routing Basics (Continuity Equation) A reservoir has an initial storage of $100,000 \text{ m}^3$ at $t=0$. During the next 2-hour period ($\Delta t = 2 \text{ hours}$), the average inflow ($I_{avg}$) is $15 \text{ m}^3/\text{s}$ and the average outflow ($O_{avg}$) over the spillway is $10 \text{ m}^3/\text{s}$. Calculate the new volume of storage ($S_2$) in the reservoir at the end of the 2-hour period.

Case Study 1: The Function of Detention Basins in Urban Environments An engineer designs a dry detention basin for a new subdivision to mitigate the effects of increased runoff from paved surfaces. Discuss how the principles of reservoir routing explain the basin's ability to protect downstream properties from flooding.

Case Study 2: Muskingum Weighting Factor ($x$) in Natural Channels The Muskingum method uses a weighting factor, $x$, which varies between 0 and 0.5. Discuss the physical significance of this parameter in relation to "wedge storage" and how varying $x$ changes the shape of a routed flood wave.