Problem:

In a given year, a watershed with an area of 2500 km² received 130 cm of precipitation. The average rate of flow measured in a gage at the outlet of the watershed was 30 m³/sec. Estimate the water losses due to the combined effects of evaporation, transpiration, and infiltration due to ground water. How much runoff reached the river for the year (in cm)?

Solution:

The following are given:

\begin{align*}
\text{Area}, A & = 2500 \ \text{km}^2\\
\text{Precipitation}, P & = 130\ \text{cm} \\
\text{Outflow}, O & = 30 \ \text{m}^3/\text{s}
\end{align*}

We know that in a watershed,

\text{inflow} - \text{water losses} = \text{outflow}

The inflow is the precipitation.

For this case, the outflow O is also the runoff, R. We will convert the runoff R in the unit of cm by dividing the given R by the area, and multiplying it by 1 year.

\begin{align*}
R & = \frac{30 \ \frac{\text{m}^3}{s} \times \frac{100\ \text{cm}}{1\ \text{m}} \times \frac{3600\ \text{s}}{1\ \text{hr}} \times \frac{24\ \text{hr}}{1\ \text{day}}\times \frac{365\ \text{days}}{1\ \text{year}} \times 1 \ \text{year}}{2500\ \text{km}^2\times \left( \frac{1000\ \text{m}}{1\ \text{km}} \right)^2} \\
R & = 37.8432\ \text{cm}

\end{align*}

We can now compute for the water losses.

\begin{align*}
\text{water losses} & = \text{inflow} - \text{outflow} \\
\text{water losses} & = 130\ \text{cm} - 37.8432\ \text{cm} \\
\text{water losses} & = 92.1568\ \text{cm}
\end{align*} 

Therefore, the combined water losses is \boxed{92.1568\ \text{cm}[\katex] and the runoff for the river for the year is [katex]\boxed{37.8432\ \text{cm}.