$.$ An organic chemical is produced by a batch process. In this process, chemicals $x$

and $Y$ react to form chemical Z$.$ Since the reaction rate is very high, the total time

required per batch has been found to be independent of the amounts of the

materials, and each batch requires $2h,$ including time for charging, heating, and

dumping. The following equation shows the relation between the pounds of $Z$

produced $($lb$,)$ and the pounds of $,)$ and $,)$ supplied:

$lb,=1\mathrm{.}5(1\mathrm{.}1lb,lbz+1\mathrm{.}3lb,lb,-l{b}_{x}l{b}_Y{)}^{0\mathrm{.}5}$

Chemical X costs $$0\mathrm{.}09$ per pound. Chemical Y costs $$0\mathrm{.}04$ per pound. Chemical Z

sells for $$0\mathrm{.}80$ per pound. If one$-$half of the selling price for chemical $Z$ is due to

costs other than for raw materials, what is the maximum profit obtainable per pound

of chemical Z$?$