Chapter 5.6, Problem 37PS

Solution

To calculate: Find the value of golden ratio.

The value of golden ratio is $\frac{1+\sqrt{5}}{2}$.

Given information:

  $\frac{w}{l}=\frac{l}{w+l}$

Concept used:

Golden ratio: If the ratio of two quantities is the same as the ratio of their sum to the bigger of the two quantities, then the two quantities are in the golden ratio.

Quadratic formula: $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Given that;

  $\frac{w}{l}=\frac{l}{w+l}$

Diagonally multiply;

  $\begin{array}{l}w\left(w+l\right)=l^2\\ w^2+wl=l^2\end{array}$

Subtract $l^2$ from both sides;

  $w^2+wl-l^2=0$

Multiply both sides by $-1$ ;

  $l^2-wl-w^2=0$

Substitute the $w=1$ ;

  $l^2-l-1=0$

Use the quadratic formula: $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  $\begin{array}{l}l=\frac{-\left(-1\right)\pm \sqrt{{\left(-1\right)}^2-4\left(1\right)\left(-1\right)}}{2\left(1\right)}\\ =\frac{1\pm \sqrt{1+4}}{2}\\ =\frac{1\pm \sqrt{5}}{2}\end{array}$

Take positive value of length $l$ ,

  $l=\frac{1+\sqrt{5}}{2}$

Conclusion:

  $l=\frac{1+\sqrt{5}}{2}$