A parabolic arch bridge is modeled by the quadratic function $h(x)$, representing the height of the arch in meters, $x$ meters horizontally from the left base. The arch has a maximum height of $15$ meters at a horizontal distance of $20$ meters from the left base. The right base of the arch is at a horizontal distance of $40$ meters from the left base, where the height is $0$. Based on this model, what is the estimated height of the arch, in meters, at a horizontal distance of $10$ meters from the left base?