Analytically solving Tarzan's dilemma

Teachers who use the instructive Tarzan exercise in the November 2013 issue¹ may find it useful to know that TPT previously published an analytic solution² for the maximum range. To review the problem, Tarzan grabs the end of a vertically hanging vine of length L at height h above the ground with initial speed v. He swings up to angle θ and then lets go of the vine, landing on the ground a horizontal distance R away from the point at which he grabbed the vine. Normalizing the variables by defining $w≡v2/2gL$ and $y≡h/L$⁠, then the angle $θmax⁡$ at which Tarzan should release the vine to maximize his range is given by

where...