Company headquarters holds $100$ units of a raw material; it proposes to divide this into three lots of $x,y$ and $z$ units of raw material to be sent to three area factories in the North, the Midlands and the South of the country and there they are turned into a finished product. Although the product sells at the same price in all three regions, the factories are not equally productive and in fact their respective outputs are:

North: $3\sqrt[\phantom{2}]{x}$

Midlands: $4\sqrt[\phantom{2}]{y}$

South: $5\sqrt[\phantom{2}]{z}$

North: $3\sqrt[\phantom{2}]{x}$

Midlands: $4\sqrt[\phantom{2}]{y}$

South: $5\sqrt[\phantom{2}]{z}.$

How should the headquarters allocate $x,y,z$ to maximise the revenue? $[$Hint: Use the Lagrange multiplier method.$]$