The XYZ Company produces two products. The total profit achieved from these products $($in thousands of dollars$)$ is described by the following equation.

Total profit $=-0\mathrm{.}3$ x$\_1^2-0\mathrm{.}5$ x$\_2^2+7$ x$\_1+11$ x$\_2+1,500$

where

$[$ x$\_1=$ thousands of units of product $1$; x$\_2=$ thousands of units of product $2]$

Every $1,000$ units of product $1$ require $1$ hour of time in the shipping department, and every $1,000$ units of product $2$ require $30$ minutes in the shipping department. Each unit of each product requires two pounds of a special ingredient, of which $64,000$ pounds are available. Additionally, $80$ hours of shipping labor are available. Demand for X$\_1$ and X$\_2$ is unlimited. $($a$)$ Formulate an NLP model for this problem to maximize profit $($in thousands of dollars$).$ MAX: Subject to: shipping department special ingredient

x$\_1,$ x$\_2>=0$

$($b$)$ Implement your model in a spreadsheet and solve it$.$ What is the optimal solution? $($Round your numerical values to three decimal places.$)$

$($x$\_1,$ x$\_2)=($