Answer
The minimum distance is 10.8 feet
Work Step by Step
We can convert the angle to degrees:
$\theta = 23^{\circ}20' = (23+\frac{20}{60})^{\circ} = 23.3^{\circ}$
We can find the minimum distance $x$:
$\frac{4.65~ft}{x} = tan(23.3^{\circ})$
$x = \frac{4.65~ft}{tan(23.3^{\circ})}$
$x = 10.8~ft$
The minimum distance is 10.8 feet