Question

Now there is a door whose height is more than its width by 6 chi 8 cun. The distance between the [opposite] corners is 1 zhang. Find the height and width of the door.NOTE: 1 zhang = 10 chi = 100 cun. Give your answers inunits of chi and cun.

Answer 1

Height: 9 chi 6 cun

Width: 2 chi 8 cun

Explanation:

Now there is a door whose height is more than its width by 6 chi 8 cun. The distance between the [opposite] corners is 1 zhang. The height of the door is 9 chi 6 cun and the width of the door is 2 chi 8 cun.

1 zhang = 10 chi = 100 cun

Answer 2

• height: 9 chi 6 cun
• width: 2 chi 8 cun

Explanation:

The factor-of-ten relationship between the different units means we can combine the numbers in decimal fashion. If 1 unit is 1 zhang, then 1 chi is 0.1 zhang and 1 cun is 0.01 zhang. Hence 6 chi 8 cun is 0.68 zhang.

Let x and y represent the width and height, respectively. In terms of zhang, we have ...

y - x = 0.68

x^2 +y^2 = 1^2

Substituting y = 0.68 +x into the second equation gives ...

x^2 + (x +0.68)^2 = 1

2x^2 +1.36x - 0.5376 = 0 . . . . . eliminate parentheses, subtract 1

Using the quadratic formula, we have ...

x = (-1.36 ±√(1.36^2 -4(2)(-0.5376)))/(2·2) = (-1.36 ±√6.1504)/4

x = 0.28 . . . . . the negative root is of no interest

y = 0.28 +0.68 = 0.96

In units of chi and cun, the dimensions are ...

height: 9 chi 6 cun

width: 2 chi 8 cun