Question $1$

$(15$ points$)$ A manufacturing facility produces two products, A and B$.$ The products consume resources and generate profit as shown in the table below:

$\backslash$table$[[$Resource$,$A Product Consumption,B Product C$$nsumption$,$Total Resource Amount$],[$Resource $1,2$ units,$3$ units,$100$ units$],[$Resource $2,4$ units,$2$ units,$80$ units$],[$Resource $3,3$ units,$5$ units,$120$ units$]]$

Additionally, the profit per unit for the products is as follows:

A product: $30TL$

B product: $40TL$

The facility must produce each product at a certain quality level. The quality of the products is related to the resource consumption rates. Formulate and solve a Linear Programming $($LP$)$ model to maximize the total profit while ensuring the quality requirements.