A catchment may be idealized as a circle of radius 30 km. There are five rain gauges, one at the center of the catchment and four on the boundary (equi-spaced), as shown in the figure (not to scale). The annual rainfall recorded at these gauges in a particular year are given below.
Using the Thiessen polygon method, what is the average rainfall (in mm, rounded off to two decimal places) over the catchment in that year?
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To calculate the average rainfall using the Thiessen polygon method, simply take the weighted average where each rain gauge is weighted equally, as they represent equal portions of the catchment area.
To calculate the average rainfall using the Thiessen polygon method, the areas corresponding to the gauges are assumed equal because they are placed at equidistant points on the boundary. Since the total area is divided into five parts, each gauge covers \( \frac{1}{5} \) of the total area. Thus, the average rainfall can be calculated using the following weighted average formula:
\[
\text{Average Rainfall} = \frac{1}{5} (R_1 + R_2 + R_3 + R_4 + R_5)
\]
Substitute the values:
\[
\text{Average Rainfall} = \frac{1}{5} (910 + 930 + 925 + 895 + 905)
\]
\[
\text{Average Rainfall} = \frac{1}{5} (4565) = 912.28 \, \text{mm}
\]
Thus, the average rainfall over the catchment in that year is approximately 912.28 mm (rounded to two decimal places).
