2240.11 – A Butterfly Garden

At a nearby retirement home, a large garden (for butterflies) is being planned. It will be $40 \ \ft \times 60 \ \ft$ , and will be crossed by four straight smooth paths, all the same width, as seen in the figure below. Turning this into a puzzle for you, if the total area of the paths is $736 \ \ft^2$ , how wide is each path?

Solution

If $x$ is the width of the paths (as in the figure below) then we can compute the area as follows:

$736 \ \ft^2 = 2 x \cdot 60 + 2x \cdot 40 - 4x^2$

Here the term $2x \cdot 60$ is the area of the two horizontal paths, $2x \cdot 40$ similarly is the area of the vertical paths, and because the square patches where the paths cross are now each counted twice we must subtract $4 x^2$ . From the equation above we derive the quadratic:

$\begin{align*}0 &= 4x^2 - 200x + 736 \\&= 4 (x - 4) (x - 46)\end{align*}$

Discarding the irrelevant root $46$ , we find that $x = 4 \ \ft$ .

This is certainly wide enough for one wheelchair, but perhaps not for two unless they take turns going, or unless one-way signs are posted.