Question

1. if a golden rectangle has a width of 9 cm, what is its length?

Explanation:

Step1: Recall golden - ratio formula

The ratio of the length $l$ to the width $w$ of a golden rectangle is the golden ratio $\varphi=\frac{1 + \sqrt{5}}{2}\approx1.618$. So, $\frac{l}{w}=\frac{1+\sqrt{5}}{2}$.

Step2: Solve for length $l$

We know $w = 9$ cm. Rearranging the formula $\frac{l}{w}=\frac{1+\sqrt{5}}{2}$ gives $l=w\times\frac{1 + \sqrt{5}}{2}$. Substitute $w = 9$ into the formula: $l=9\times\frac{1+\sqrt{5}}{2}$.
Calculate $l = 9\times\frac{1 + 2.236}{2}$ (since $\sqrt{5}\approx2.236$). First, $1+2.236 = 3.236$. Then, $\frac{3.236}{2}=1.618$. Finally, $l=9\times1.618 = 14.562$ cm.

Answer:

$14.562$ cm