A 'hole' new dimension

A small hole is made in the side of a full oil drum (barrel). A jet of oil shoots from the drum and lands on the ground some distance from the drum. At what height (in metres) above the ground should the hole be made to maximise the horizontal distance between the initial splash down point and the drum? Give your answer to three decimal places.

Note. It is not too hard to find the answer using standard formulas from mechanics and hydrodynamics. My challenge is to find the answer from first principles using dimensional analysis without appealing to a formula sheet (not even a memorised one!).

Assumptions

\(\text{The drum is 1 metre tall and open to the atmosphere at the top.}\\\text{acceleration due to gravity }g=10ms^{-2} \\ \text{density of oil }\rho=900 kgm^{-3}\)

Air resistance to the jet of oil may be ignored.

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