A tennis ball launcher is located at x = 0 on top of a hill whose height z(x) is described by $z = 4h − (x^2)/h$, where h is a constant.

Question

The machine launches balls (horizontally) at speed $V = \frac{\sqrt{gh}}{2}$.

(a) Find the value of x for the most distant point on the hillside that the gun can hit.

(b) A target is located on the hillside at x = 2h. Determine the smallest value of x which to which the machine must be moved so that the target is just within range.

Now I've already managed to do part (a) and have $x_{max} = \frac{\sqrt{186}}{12}h$.

Assuming my solution to part (a) is correct how do I go about finding the solution to part (b)?

Many thanks.