Answer
a. $\frac{1}{\Phi-1}$
b. $\Phi=\frac{1+\sqrt {5}}{2}$
c. $\frac{1+\sqrt {5}}{2}$ to 1.
Work Step by Step
a. The ratio for rectangle B is given by $\frac{1}{\Phi-1}$ that is the longer side divided by the shorter side.
b. Let $\frac{\Phi}{1}=\frac{1}{\Phi-1}$, multiply both side with $\Phi-1$, we have $\Phi(\Phi-1)=1$ or $\Phi^2-\Phi-1=0$. Use the formula for a quadratic equation, we have $\Phi=\frac{1\pm\sqrt {1+4}}{2}$. As $\Phi\gt0$, we have $\Phi=\frac{1+\sqrt {5}}{2}$
c. Based on the above result, we can conclude that: The ratio of the long side to the short side in a golden rectangle of any size is $\frac{1+\sqrt {5}}{2}$ to 1.
