A chemical manufacturing plant can produce $z$ units of chemical Z given $p$ units of chemical P and $r$ units of chemical R$,$ where:

$z=30p^{0\mathrm{.}84}{r}^{0\mathrm{.}16}$

Chemical P costs $$200$ a unit and chemical $R$ costs $$1,200$ a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $$300,000.$

A$)$ How many units each chemical $(P$ and $R)$ should be "purchased" to maximize production of chemical $Z$ subject to the budgetary constraint?

Units of chemical $P,p=$

Units of chemical $R,r=$

B$)$ What is the maximum number of units of chemical Z under the given budgetary conditions? $($Round your answer to the nearest whole unit.$)$

Max production, $z=$ units