Answer:
37.2
Explanation:
A golden rectangle is a rectangle that satisfies the following proportion:
[tex]\frac{a+b}{a}=\frac{a}{b}[/tex]Where a is the shortest side and (a+b) is the length of the other side. So, we can replace a by 23 and calculate b as:
[tex]\frac{23+b}{23}=\frac{23}{b}[/tex]Applying cross-multiplication, we get:
[tex]\begin{gathered} (23+b)\cdot b=23\cdot23 \\ 23b+b^2=529 \\ b^2+23b-529=0 \end{gathered}[/tex]Therefore, we can find the solutions to the quadratic function as:
[tex]\begin{gathered} b=\frac{-23\pm\sqrt[]{23^2-(4\cdot1\cdot(-529))}_{}}{2\cdot1} \\ b=14.215 \\ or \\ b=-37.215 \end{gathered}[/tex]Since -37.215 doesn't have any sense here, the value of b is 14.215
It means that the length of the longest side is:
a + b = 23 + 14.215 = 37.215
So, the answer is 37.2
