The Cobb$-$Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form

f$($L$,$ C$)=$ c$0$Lc$1$Cc$2$

where

c$0,$ c$1,$ and c$2$

are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital.

$($a$)$

In this example, assume

c$0=5,$ c$1=0\mathrm{.}25,$ and c$2=0\mathrm{.}75.$

Assume each unit of labor costs $$25$ and each unit of capital costs $$75.$ With $$60,000$ available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output.

Max$$

s$.$t$.$

L$,$ C $0$

$($b$)$

Find the optimal solution to the model you formulated in part $($a$).$ What is the optimal solution value $($in units$)?($Hint: When using Excel Solver, use the bounds

$0$ L $3,000$

and

$0$ C $1,000.$

Round your answers to the nearest integer when necessary.$)$

units at

$($L$,$ C$)=$

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 $+9$

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 $9,000$ 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$.$

X$,$ Y $0$

$($b$)$

Solve the optimization model to determine the optimal number of gloves to produce at each facility. What is the optimal solution value $($in dollars$)?($Round the cost to the nearest integer and Xand Y to two decimal places.$)$

$ at

$($X$,$ Y$)=$