We will use notation \sqrt{^{n}\;x}\quad for n nested square roots (see tehtmi 's answer): \sqrt{^{n}\;x} = \underbrace{\sqrt{\sqrt{\cdots \sqrt{\sqrt{x}}}}}_{n} = \sqrt[2^n]{x}. Here is one ...

You've made mistakes for the components of the initial velocity. \displaystyle u_{x}=v\cos 45^{\circ}=\frac{v}{\sqrt{2}} and \displaystyle u_{y}=v\sin 45^{\circ}=\frac{v}{\sqrt{2}}

Since we have (x-20)^2+(z-40)^2=100^2-25^2 x=-\sqrt 3z-5\sqrt 2 we have (-\sqrt 3z-5\sqrt 2-20)^2+(z-40)^2=100^2-25^2. Expand these to get the form Az^2+Bz+C=0 and use z=\frac{-B\pm\sqrt{B^2-4AC}}{2A}. ...