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

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

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 $5,000$ 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 facility.

min

s$.$t$.$

$=5$

$x,Y0$

$($b$)$ Use Excel Solver or LINGO to find the solution to your mathematical model to determine the optimal number of gloves to produce at

each facility. What is the optimal solution value $($in dollars$)?$

$

$at(x,Y)=(,)$