retired couple supplement their income by making fruit pies, which they sell to a local grocery store. During the month of September, they produce apple and grape pies. The apple pies are sold for $$4\mathrm{.}50$ to the grocer, and the grape pies are sold for $$3\mathrm{.}60.$ The couple is able to sell all of the pies they produce owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month. For the month of September, they have $1,200$ cups of sugar and $2,100$ cups of flour. Each apple pie requires $1(1)/(2)$ cups of sugar and $3$ cups of flour, and each grape pie requires $2$ cups of sugar and $3$ cups of flour.

a$.$ Determine the number of grape and the number of apple pies that will maximize revenues if the couple working together can make an apple pie in six minutes and a grape pie in three minutes. They plan to work no more than $60$ hours.

b$.$ Determine the amounts of sugar, flour, and time that will be unused. $($Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$