Modern Furniture is a manufacturer that produces two types of tables made from Oak wood and just received this month's allocation of $2500$ planks $($b$)$ of wood.

It takes $5$ planks to make a unit of product $1,$ and $3$ planks of woods to make a unit of product $2.$

Product $1$ generated $90$USD profit $($p$1),$ while product $2$ generated $40$USD $($p$2).$

Modern Furniture has a contract calling for at least $50$ units $($m$)$ of product $1$ this month.

The firm's facilities are such that at most $500($u$)$ units of product $2$ may be produced monthly.

This month's production level of product $1$ is $($x$1)$ and product $2$ is $($x$2),$ respectively.

Question:

Write mathematical model summary and find the optimal solution with the maximum profit $($in USD$)$