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Explore the broken bamboo problem by varying the height (

hb+c

) of the stalk; horizontal offset

a

of the tip of the broken, tilted stalk; and counterclockwise angle by which the stalk is bent from its original vertical position to an angle at which it just touches the ground:

​
h a bend angle

Computational Explanation

To solve the broken bamboo problem, first draw a figure (with arbitrary dimensions for now) showing the relationship between the original bamboo height

hb+c

, horizontal offset of the broken tip

a

and height of the break

b

:

TriangleDiagram[a_,b_,c_,t_:0]:=With{dx=3.,h=b+c,θ=ArcTan[b,a]},cs[x_]:={Cos[x],Sin[x]};GraphicsColorData[97,5],Circle{0,b},0.5,θ-

π

2

,

π

2

-t,Line[{{0,1},{1,1},{1,0}}0.3],AbsoluteThickness[2],ColorData[97,3],AbsoluteDashing[5],Line{0,b},{0,b}+ccs

π

2

-t,Line[{{0,b},{a,0}}],ColorData[97,4],Line[{{0,b},{0,0},{a,0}}],GrayLevel[0,1],Text"c",{0,b}+.5ccs

π

2

-t,-dxcs[π-t],Text"b",0,

b

2

,{dx,0},Text"a",

a

2

,0,{0,dx},Text"c",

{a,b}

2

,-dxcs[θ]

In[]:=

TriangleDiagram[3,4,5]

Out[]=

In the figure, the dashed line corresponds to the portion of the bamboo stalk that is bent in a counterclockwise direction until it touches the ground level with its base. Both the dashed line and the hypotenuse of the right triangle therefore have the same length

c

.

Use the Pythagorean theorem to relate the sides and hypotenuse of the right triangle pictured above and solve for the unknown height of the break

b

:

In[]:=

Solve[b^2+a^2(h-b)^2,b]

Out[]=

b

-

2

a

+

2

h

2h



Plugging back in the values from the original expression then gives:

In[]:=

Solve[b^2+a^2(h-b)^2,b]/.{a3,h10}

Out[]=

b

91

20



Expressed as a mixed fraction, the height of the break is therefore, as correctly given in the original, 4 + 11/20:

In[]:=

MixedFractionParts

91

20



Out[]=

4,

11

20



Finally, plug in all values to obtain a correctly dimensioned solution of the problem:

TriangleDiagram[a,b,h-b]/.{a3,b91/20,h10}

Out[]=

Other Resources

Additional Reading

• Dauben, J. W. "Chinese Mathematics." Ch. 3 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 284 and 287–288, 2007.
• Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 257–258, 2011.
• Lam, L. Y. "On the Existing Fragments of Yang Hui's Hsiang Chieh Suan Fa." Archive for History of Exact Sciences, Vol. 6, pp. 82–88, 1969.
• Lam, L. Y. A Critical Study of the Yang Hi Suan Fa, a Thirtheenth-Century Chinese Mathematical Treatise. Singapore: Singapore University Press, 1977.
• Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 64–66, 2007.
• Mankiewicz, R. The Story of Mathematics. Princeton, NJ: Princeton University Press, pp. 34–36, 2004.
• Martzloff, J.-C. A History of Chinese Mathematics (Trans. S. S. Wilson). New York: Springer-Verlag, pp. 300–301, 1997.
• Needham, J. Science and Civilization in China, Vol. 3. Cambridge, England: Cambridge University Press, 1959.

Additional Links

• Wikipedia: The Nine Chapters on the Mathematical Art
• Wikipedia: Pythagorean theorem
• Wikipedia: Yang Hui