= Solution
At the cone's vertex height $h$, the first part gives $U(h)=U_0\sqrt{1-2gh/U_0^2}$ and $A(h)U(h)=A_0U_0$. Insert these into the assumed local impingement force from part (ii). Static support requires upward force equal to $Mg$, giving
$$
\boxed{\rho A_0U_0^2\sqrt{1-\frac{2gh}{U_0^2}}(1-\cos\alpha)=Mg.}
$$
Equivalently,
$$
h=\frac{U_0^2}{2g}\left[1-\left(\frac{Mg}{\rho A_0U_0^2(1-\cos\alpha)}\right)^2\right].
$$
For a nonnegative support height the ratio in parentheses must not exceed one, and the prescribed $h\ll U_0^2/(2g)$ requires that ratio to be close to one. If the weight exceeds the lift available at the nozzle, this model has no supported height above the nozzle.
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