$.$ A small brewery reuses its own empty bottles, but it does not have a bottle$-$cleaning machine. A delivery truck collects the empty bottles from liquor stores and bars and brings them back to the brewery. Once enough empty bottles accumulate, the truck transports them to a cleaning facility and brings back clean bottles. The cleaning facility charges $11$ cents to wash a bottle, and cost of trucking is $2$ cents per bottle. Management is thinking of buying an automated bottle$-$cleaning machine. The machine costs $$1\mathrm{.}5$ million and has a useful life of $10$ years. The brewery will use the machine only when enough empty bottles accumulate. The operating cost of the machine will be $$108$ per hour, and it can wash $60$ bottles per minute.

a$.$ At what quantity of bottles would total annual cost of cleaning in$-$house equal total annual cost of outsourcing the cleaning?