Hachiman Wake Shrine is located 4.6 km northwest of Kumayama Station on the JR Sanyo Main Line.
Problem
There are three sections (of different heights), upper, middle and lower, in the rice field. The areas of the upper, middle and lower sections are 210, 150 and 240$m^2$ respectively. The height differences between the upper and lower sections, the upper and middle sections, and the middle and lower sections are 2.5, 1.5 and 1.0$m$ respectively. When 3 sections are leveled to make one section with the same height, how much are the upper, middle and lower sections to be leveled?
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Solution
Assuming that the height of the lower section is 0 m, the volume of soil that makes up the upper section is
$$210×2.5=525 m^3.$$
Also, the volume of the middle section is
$$150×1=150m^3.$$
Therefore, the total volume of the three sections is
$$V_1=525+150=675m^3.$$
Letting $x$ $m$ be the level difference between the leveled field and the lower section, the total volume of the large field is
$$V_2=(210+150+240)×x=600x.$$
Since $V_1=V_2$, we have
$$600x=675.$$
$$∴ x=1.125 m.$$
The height of the middle section is $1m$, so the height difference from the leveled field is
$$y=1.125-1=0.125 m.$$
Since the height of the upper section is 2.5m, the height difference from the leveled field is
$$z=2.5-1.125=1.375 m.$$
(Answer) The upper section will be excavated $1.375 m$, and the middle and
lower sections will be reclaimed $0.125$ and $1.125 m$ respectively.
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.49; p.313.