Henry Ernest Dudeney/Puzzles and Curious Problems/208 - Dividing the Board/Solution

by : $208$

 * Dividing the Board

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 $5.8114$.