160
Chapter 5
Figure 5.20. Find the semimajor axis of an ellipse in terms of the radius of the circle when the area of each ellipse is maximized.
Answer: a = r / 2 . Example: If r = 99, then a = 70.0035. (Kobayashi had recognized 1.414215004.)
2 = 99/70.0035 ≅
Turn to page 181 for a full solution.
Problem 22 This problem, proposed by Kuno Hirotomo, can be seen as the third from the right on the second Atsuta tablet, color plate 12. Hung in 1844 by Takeuchi Shu ¯ kei (1815–1873), the tablet was lost. More recently, the shrine constructed a replica based on the anonymous and undated note, “Sandai Gakumen Sya,” or “Record of Sangaku.” Two parallel planes, separated by a distance d, cut a sphere of radius r (see figure 5.21). Find the surface area S of the cut- out section in terms of d and r. Answer: S = 2πrd. A solution can be found on page 182.
d
Figure 5.21. Find the surface area of the slice (black) in terms of d and the radius of the sphere.