Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment, capacity, and so on$.$ The Dayton plant has weekly costs that can be expressed as a function of the number of gloves produced

TCD$($X$)=$ X$2$ X $+7$

where X is the weekly production volume in thousands of units and

TCD$($X$)$

is the cost in thousands of dollars. The Hamilton plant's weekly production costs are given by

TCH$($Y$)=$ Y$2+2$Y $+8$

where Y is the weekly production volume in thousands of units and

TCH$($Y$)$

is the cost in thousands of dollars. Heller Manufacturing would like to produce $7,000$ gloves per week at the lowest possible cost.

$($a$)$

Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facil