Chiu Chang Suann Jing/Examples/Example 8
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Example of Problem from Chiu Chang Suann Jing
- There is a bamboo $10$ feet high,
- the upper end of which being broken down on reaching the ground,
- the tip is just $3$ feet from the stem;
- what is the height of the break?
Solution
The height of the break is $4 \tfrac {11} {20}$ feet.
Proof
Let the height of the break be $x$ feet.
The length of the broken section is then $10 - x$ feet.
The broken section forms the hypotenuse of a right triangle.
One of the legs of that right triangle is the remaining stalk of bamboo, which is $x$ feet long.
The other leg is the distance of the end of the tip from the stalk, which is $3$ feet.
Hence:
\(\ds \paren {10 - x}^2\) | \(=\) | \(\ds x^2 + 3^2\) | Pythagoras's Theorem | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 20 x\) | \(=\) | \(\ds 100 - 9\) | simplification | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \frac {91} {20}\) | simplification | ||||||||||
\(\ds \) | \(=\) | \(\ds 4 \tfrac {11} {20}\) | simplification |
$\blacksquare$
Historical Note
This problem was also presented by Brahmagupta in one of his works, but it has not been identified where this appears.
Sources
- c. 100: Anonymous: Chiu Chang Suann Jing
- 1965: Henrietta Midonick: The Treasury of Mathematics: Volume $\text { 1 }$
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Nine Chapters: $66$