Chapter 9.1, Problem 106E

a.

To determine

To show that both golden ratio and golden ratio conjugate are both solution of $x^2-x-1=0$ .

Answer to Problem 106E

Golden ratio is $\frac{1+\sqrt{5}}{2}$

Golden ratio conjugate is $\frac{1-\sqrt{5}}{2}$

Equation is $x^2-x-1=0$

Calculation:

Golden ratio

  $0=0$

Golden ratio conjugate

  $0=0$

Explanation of Solution

Given information:

Golden ratio is $\frac{1+\sqrt{5}}{2}$

Golden ratio conjugate is $\frac{1-\sqrt{5}}{2}$

Equation is $x^2-x-1=0$

Calculation:

Golden ratio $=\frac{1+\sqrt{5}}{2}$

Equation

  ${\left(\frac{1+\sqrt{5}}{2}\right)}^2-\left(\frac{1+\sqrt{5}}{2}\right)-1=0$

On solving

  $\left(\frac{1+2\sqrt{5}+5}{4}\right)-\left(\frac{1+\sqrt{5}}{2}\right)-1=0$

On solving

  $\left(\frac{1+2\sqrt{5}+5}{4}\right)-\left(\frac{2+2\sqrt{5}}{4}\right)-\frac{4}{4}=0$

On solving

  $\left(\frac{1+2\sqrt{5}+5-2-2\sqrt{5}-4}{4}\right)=0$

On solving

  $\left(\frac{0}{4}\right)=0$

On solving

  $0=0$

Hence proved

Golden ratio conjugate $=\frac{1-\sqrt{5}}{2}$

Equation

  ${\left(\frac{1-\sqrt{5}}{2}\right)}^2-\left(\frac{1-\sqrt{5}}{2}\right)-1=0$

On solving

  $\left(\frac{1-2\sqrt{5}+5}{4}\right)-\left(\frac{1-\sqrt{5}}{2}\right)-1=0$

On solving

  $\left(\frac{1-2\sqrt{5}+5}{4}\right)-\left(\frac{2-2\sqrt{5}}{4}\right)-\frac{4}{4}=0$

On solving

  $\left(\frac{1-2\sqrt{5}+5-2+2\sqrt{5}-4}{4}\right)=0$

On solving

  $\left(\frac{0}{4}\right)=0$

On solving

  $0=0$

Hence proved

b.

To determine

To constract a diagram show golden ratio.

Answer to Problem 106E

  

Explanation of Solution

Given information:

Constract diagram show golden ratio

Pentagram represent golden ratio and which is approximately $1.618$

Here lengths of pentagram are

  $a=216$

  $b=133$

  $c=82$

  $d=51$

Golden ratios of lengths

  $\frac{a}{b}=\frac{216}{133}=1.624$

  $\frac{b}{c}=\frac{133}{82}=1.622$

  $\frac{c}{d}=\frac{82}{51}=1.618$