Chapter 10.3, Problem 27P

The length of a golden rectangle is $4+4\sqrt{5}$.

Use the ratio of length to width $\left(1+\sqrt{5}\right):2$ to find the width of the golden rectangle.

To determine

To find: the width of the golden rectangle.

Answer to Problem 27P

The width of the golden rectangle is $8$ .

Explanation of Solution

Given information:

The length of the golden rectangle is $4+4\sqrt{5}$ and length to width ratio is $\frac{1+\sqrt{5}}{2}$ .

Calculation: to find the width of the golden rectangle, first use the ratio and then simplify,

Let $w$ is the width of the golden rectangle,

  $\begin{array}{l}\frac{1+\sqrt{5}}{2}=\frac{4+4\sqrt{5}}{w}\\ \frac{1+\sqrt{5}}{2}=\frac{4\left(1+\sqrt{5}\right)}{w}\text{ [simplify]}\\ \frac{1}{2}=\frac{4}{w}\text{ [divide out common factor]}\\ w=8\text{ [cross multiplication]}\end{array}$

Thus, The width of the golden rectangle is $8$ .