Question

Susie wants to make a quilt in the shape of a golden rectangle. She only has enough binding material for a quilt with a perimeter of 31 feet. What should the dimensions of the quilt be?

Answer 1

A golden rectangle has the following dimensions:

which satisfy
\[\frac{a+b}{a}=\frac{a}{b}\approx1.618\]
The perimeter of such a rectangle will be
\[P=2a+2(a+b)=4a+2b\]
Cross-multiplying in the ratio equation, you get
\[(a+b)b=a^2~~\iff~~a^2-ab-b^2=0\]
Solve for one of the variables in the perimeter equation:
\[4a+2b=31~~\iff~~\begin{cases}a=\dfrac{31-2b}{4}&\text{or}\\\\
b=\dfrac{31-4a}{2}\end{cases}\]
then substitute into the modified ratio equation:
\[\left(\frac{31-2b}{4}\right)^2-\left(\frac{31-2b}{4}\right)b-b^2=0\\~~~~~~~~~~~~~~~~~~~~~~~~\text{or}~~\\
a^2-a\left(\frac{31-4a}{2}\right)-\left(\frac{31-4a}{2}\right)^2=0\]
Whichever path you take the answer will be the same.