Henry Ernest Dudeney/Puzzles and Curious Problems/208 - Dividing the Board/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $208$
- Dividing the Board
- A man had a board measuring $10$ feet in length, $6$ inches wide at one end, and $12$ inches wide at the other,
- as shown in the diagram.
- How far from $B$ must the straight cut at $A$ be made in order to divide it into two equal pieces?
Solution
Approximately $5.8114$ feet from $B$.
Proof
Let $d$ be the length of the cut at $A$ from $B$.
The length of this cut is $6 + \dfrac {6 d} {10}$.
From Area of Trapezoid we have (indirectly) that:
- $\paren {12 + 6 + \dfrac {6 d} {10} } \paren {10 - d} = \paren {6 + 6 + \dfrac {6 d} {10} } d$
After simplification, this gives:
- $d^2 + 20 d - 150 = 0$
Using the Quadratic Formula, this gives:
- $d = -10 \pm \sqrt {250}$
which has a positive root approximately equal to $5.8114$.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $208$. -- Dividing the Board
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $272$. Dividing the Board