91Ó°ÊÓ

Use the Chinese square root algorithm to find the square root of 142,884 .

Solve problem 28 of chapter 6 of the Nine Chapters: A man is carrying rice on a journey. He passes through three customs stations. At the first, he gives up \(1 / 3\) of his rice, at the second \(1 / 5\) of what was left, and at the third, \(1 / 7\) of what remains. After passing through all three customs stations, he has left 5 pounds of rice. How much did he have when he started? (Versions of this problem occur in later sources in various civilizations.)

Use calculus to confirm that the volume of the box-lid, the intersection of two perpendicular cylinders of radius \(r\), is \(\frac{16}{3} r^{3}\)

Solve problem 1 of chapter 7 of the Nine Chapters using the method of surplus and deficiency: Several people purchasein common one item. If each person paid 8 coins, the surplus is 3 ; if each paid 7, the deficiency is 4 . How many people were there and what is the price of the item?

Show that the diameter \(D\) of the largest circle that can be inscribed in a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is given by \(D=2 a b /(a+b+c)\). (This is a generalization of problem 16 of chapter 9 of the Nine Chapters, which uses the specific \(8-15-17\) triangle.)

Solve problem 24 of chapter 9 of the Nine Chapters. (This is an example of the type of elementary surveying problem that stimulated Liu Hui to write his Sea Island Mathematical Manual.) A deep well 5 feet in diameter is of unknown. depth (to the water level). If a 5 -foot post is erected at the edge of the well, the line of sight from the top of the post to the edge of the water surface below will pass through a point \(0.4\) feet from the lip of the well below the post. What is the depth of the well?

Use Qin's method to solve the pure cubic equation \(x^{3}=\) \(12,812,904\). Compare this method with the old cube root algorithm discussed in the text. In each case, show where

Solve Problem \(\mathrm{I}, 4\), from the Shushu jiuzhang, which s equivalent to \(N \equiv 0(\bmod 11), N \equiv 0(\bmod 5), N \equiv 4\) \(\bmod 9), N \equiv 6(\bmod 8), N \equiv 0(\bmod 7)\)