2230.22 – The Golden Ratio

You see here a rectangle with a special shape. The shape has the property that if you make a square at the bottom and cut it off, the remaining rectangle is similar to the original rectangle. That is,

lx=xx−l\dfrac{l}{x} = \dfrac{x}{x - l}

Find the ratio of xx to ll. This is a famous number, known as the golden ratio. Rectangles of this shape are frequently used in architecture.

Solution

First,

lx=xl−xl(l−x)=x2l2–lx–x2=0\begin{align*}\dfrac{l}{x} &= \dfrac{x}{l - x} \\[1em]l(l - x) &= {x^2} \\[0.5em]l^2 – lx – x^2 &= 0\end{align*}

We want xl\dfrac{x}{l}, so we divide by l2l^2:

1–xl−x2l2=0(xl)2+xl–1=0\begin{align*}1 – \dfrac{x}{l} - \dfrac{x^2}{l^2} &= 0 \\ \left(\dfrac{x}{l}\right)^2 + \dfrac{x}{l} – 1 &= 0\end{align*}

We then invite the quadratic formula to the show:

xl=−1±(1−(−4)2=−1±52\begin{align*}\dfrac{x}{l}&=\dfrac{-1\pm\sqrt{(1-(-4)}}{2} \\&=\dfrac{-1\pm\sqrt 5}{2}\end{align*}

We take the positive root:

xl=−1+52≈.06180\dfrac{x}{l} = \dfrac{-1 + \sqrt 5}{2} \approx .06180

Coincidentally, the ratio of 1 km1 mile≈.06214\dfrac{1\text{ km}}{1 \text{ mile}} \approx .06214.