Question

The length of a golden rectangle is 8 cm. Find the width to the nearest tenth.

Answer 1

A golden rectangle is a rectangle with sides in golden ratio

$\frac{(a+b)}{b}=\frac{b}{a}$$\begin{gathered} \text{where a}\Rightarrow is\text{ the width} \\ (a+b)\Rightarrow\text{ is the length} \end{gathered}$

From the question

$\begin{gathered} \text{lenght}=(a+b)=8 \\ a+b=8 \\ a=8-b \end{gathered}$

Substitute a in the golden ratio formula

$\begin{gathered} \frac{(a+b)}{a}=\frac{b}{a} \\ Given\text{ that the ratio of the legth to the width is }1.618 \\ \text{Thus } \\ \frac{(a+b)}{b}=1.618 \\ \text{ Since (a+b) = 8} \\ \text{Then} \\ \frac{8}{a}=1.618 \end{gathered}$

Cross multiply to find a (width)

$\begin{gathered} a(1.618)=8 \\ a=\frac{8}{1.618} \\ a=4.944 \\ a\approx4.9\operatorname{cm} \end{gathered}$

Hence, the width of the golden rectangle to the nearest tenth is 4.9cm