From the problem, we have a line segment that is 7 inches long.
It is divided into two segments (long and short) in which the ratio of long segment to the short segment is the golden ratio, Ф = 1.618
Let x = shorter segment
and the longer segment will be (7 - x)
The ratio will be :
[tex]\begin{gathered} \frac{\text{long}}{\text{short}}=\phi \\ \Rightarrow\frac{7-x}{x}=\phi \\ \Rightarrow\frac{7-x}{x}=1.618 \end{gathered}[/tex]
Solve for x :
[tex]\begin{gathered} 7-x=1.618x \\ \text{ Add x to both sides of the equation} \\ 7=1.618x+x \\ 7=2.618x \\ \text{Divide both sides by 2.618} \\ \frac{7}{2.618}=x \\ x=2.6738 \end{gathered}[/tex]
Substitute x = 2.6738 to the equation of long segment :
[tex]\begin{gathered} \text{long}=7-x \\ \Rightarrow7-2.6738 \\ \Rightarrow4.3262 \end{gathered}[/tex]
The answer rounded to 2 decimal places is 4.33 inches
