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

Puzzles and Curious Problems by Henry Ernest Dudeney: $208$

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?

Approximately $5.8114$ feet from $B$.

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 $5.8114$.

$\blacksquare$