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

where:

$z=170p^{\mathrm{.}84}{r}^{0\mathrm{.}16}$

Chemical $P$ costs $$500$ a unit and chemical $R$ costs $$1,000$ a unit. The company wants to produce as many units of

chemical $Z$ as possible with a total budget of $$250,000.$

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