Solution

First, let's find final velocity $v_f$ before she leaves the surface, at the moment she leave the surface, the normal force is zero. $N=0$

$\begin{aligned} \sum F_c &= W\cos \th - N\\ \frac{mv_f^2}{R} &= mg\cos \th - 0\\ v_f & = \sqrt{gR\cos \th} \end{aligned}$

Second, accoriding to conservation of energy,

$\begin{aligned} \frac{1}{2}mv_0^2 + mgR &= \frac{1}{2}mv_f^2 + mgH\\ 0 + mgR &= \frac{1}{2}m(\sqrt{gR\cos \th})^2 + mg \cdot R\cos\th\\ \To & \cos \th = \frac{2}{3}\\ H &= R\cos \th = 10 \cdot \frac{2}{3} \approx 6.3m \end{aligned}$