Chapter 7

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

Use the Chinese cube root algorithm to find the cube root. of \(12,812,904\).

Solve problem 3 of chapter 3 of the Nine Chapters: Three people, who have 560,350 , and 180 coins, respectively, are required to pay a total tax of 100 coins in proportion to their wealth. How much does each pay?

Solve problem 26 of chapter 6 of the Nine Chapters: There is a reservoir with five channels bringing in water. If only the first channel is open, the reservoir can be filled in \(1 / 3\) of a day. The second channel by itself will fill the reservoirin 1 day, the third channel in \(21 / 2\) days, the fourth one in 3 days, and the fifth one in 5 days. If all the channels are open together, how long will it take to fill the reservoir? (This problem is the earliest known one of this type. Similar problems appear in later Greek, Indian, and Western mathematics texts.)\\}

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?

Solve problem 8 of chapter 9 of the Nine Chapters: The height of a wall is \(10 \mathrm{ch}^{\prime} \mathrm{ih}\). A pole of unknown length leans against the wall so that its top is even with the top of the wall. If the bottom of the pole is moved 1 ch'ih farther from the wall, the pole will fall to the ground. What is the length of the pole?

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?

Find the solution to problem 3 of chapter 8 of the Nine Chapters using the Chinese method: None of the yields of 2 bundles of the best grain, 3 bundles of ordinary grain, and 4 bundles of the worst grain are sufficient to make a whole measure. If we add to the good grain 1 bundle of the ordinary, to the ordinary 1 bundle of the worst, and to the worst 1 bundle of the best, then each yield is exactly one measure. How many measures does 1 bundle of each of the three types of grain contain? Show that the solution according to the Chinese method involves the use of negative numbers.

Solve the equation \(16 x^{2}+192 x-1863.2=0\) numerically using Qin Jiushao's procedure. This equation is taken from his text.

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

Provide the details for the first step of Zhu Shijie's solution to problem 2 of his Precious Mirror. That is, let \(a\) be the base, \(b\) the altitude, and \(c\) the hypotenuse of a right triangle, and assume \(b^{2}-[c-(b-a)]=b a \quad\) and \(\quad a^{2}+c+b-a=a c\) Then set \(x=b\) and \(y=a+c\). Show that the two given equations along with the Pythagorean Theorem imply that the following two equations hold: $$ x^{3}+2 y x^{2}+2 x y-x y^{2}-2 y^{2}=0 \quad \text { and } $$ $$ x^{3}+2 y x-x y^{2}+2 y^{2}=0 $$

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

Liu Hui's method for finding the height of a distant object was used in many cultures around the world up until the seventeenth century. Curiously, this method was even used in cultures that understood methods of solving triangles using trigonometry. Discuss why this method would continue to be used, even in those circumstances.