$\cos \beta = \dfrac{1.5}{2}$
$\beta = 41.41^\circ$
$30^\circ + \beta = 71.41^\circ$
$\phi = 18.59^\circ + \alpha$
$\theta = 90^\circ - \alpha$
$\dfrac{P}{\sin 71.41^\circ} = \dfrac{2000}{\sin \phi }$
$P = \dfrac{2000 \sin 71.41^\circ}{\sin (18.59^\circ + \alpha)}$
$\dfrac{dP}{d\alpha} = \dfrac{-2000 \sin 71.41^\circ \cos (18.59^\circ + \alpha)}{\sin^2 (18.59^\circ + \alpha)} = 0$
$-2000 \sin 71.41^\circ \cos (18.59^\circ + \alpha) = 0$
$\cos (18.59^\circ + \alpha) = 0$
$18.59^\circ + \alpha = 90^\circ$
$\alpha = 71.41^\circ$ answer
$P_{min} = \dfrac{2000 \sin 71.41^\circ}{\sin (18.59^\circ + 71.41^\circ)}$
$P_{min} = 1895.65 \, \text{ lb}$ answer
$\phi = 18.59^\circ + 71.41^\circ = 90^\circ$
$\theta = 90^\circ - 71.41^\circ = 18.59^\circ$
$\dfrac{R}{\sin \theta} = \dfrac{2000}{\sin \phi}$
$\dfrac{R}{\sin 18.59^\circ} = \dfrac{2000}{\sin 90^\circ}$
$R = 637.59 \, \text{ lb}$ answer
