Answer
$w\approx 3.7$ in
Work Step by Step
Ratio of the length to the width in Golden rectangle is $\frac{1+\sqrt5}{2}$. So:
$$\frac{1+\sqrt5}{2}=\frac{6}{w}$$
$$w(1+\sqrt5)=12$$
$$w=\frac{12}{1+\sqrt5}$$
$$w=\frac{12(1-\sqrt5)}{(1+\sqrt5)(1-\sqrt5)}=\frac{12(1-\sqrt5)}{1-5}\approx3.7$$