$8\mathrm{.}2\mathrm{.}1$ Consider a $10$ km$2$ catchment receiving rainfall with intensity that varies from $0$ to $30$ mm$/$hr in a linear fashion for the first $6$ hours. Then, rainfall intensity stays constant at $25$ mm$/$hr over the next $6$ hours before it stops. During the rainfall period of $12$ hours, the rate of surface runoff from the catchment increases linearly from $0$ at t $=0$ hr to $30$ mm$/$hr at t $=12$ hr$.$

Assume that hydrologic losses, such as infiltration, evaporation, etc. are negligible in the catchment. Determine the time when the surface runoff ends, if surface runoff decreases linearly to zero from $25$ mm$/$hr at t $=12$ hr$.$

At what time does the total storage in the catchment reach its maximum and what is the corresponding storage $($in km$3)?$

Identify the times at which the rate of increase and decrease in storage are the largest.