Tamamori Shrine is located 0.4km north of Chugoku Katsuyama Station on the JR Kishin Line.
Problem
When the area enclosed by the large and small circles in the figure is 395 square inches, and the difference between the radii of the large and small circles is 5 inches, what are the diameters of the large and small circles? Note that $π=3.16$ is used in this problem.
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Solution
Let the diameter of the large circle be $x$ and the diameter of the small circle be $y$. When $k$ is the difference between the radii of the large and small circles, and $S$ is the area enclosed by the large and small circles, it follows that
$$S=π((\frac{x}{2})^2-(\frac{y}{2})^2)=π(\frac{x-y}{2})(\frac{x+y}{2}) —[1],$$
$$\frac{x-y}{2}=k —[2],$$
$$\frac{x+y}{2}=\frac{x-y}{2}+y=k+y,$$
$$\therefore S=πk(k+y) —[1]’.$$
From the problem statement,
$$S=395 —[3], k=5 —[4], π=3.16 —[5].$$
Substituting [3], [4] and [5] into [1]’, we see that
$$3.16×5×(5+y)=395,$$
$$5×(5+y)=125,$$
$$\therefore y=20 —[6].$$
From [2], [4] and [6],
$$x=y+2k=20+2×5=30.$$
(Answer) $x=30$ inches, $y=20$ inches.
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), pp.59-60; pp.285-286.