$3.$ Linear programming problem $(2\mathrm{.}5$ points$)$

A company produces two distinct products: Product A and Product B$.$ The constraints, encompassing material, machinery, and labor, are outlined as follows. Utilize the linear programming method to ascertain the optimal combination of Product A and Product B that yields the highest profit within the specified constraints. Your answer should encompass the following aspects:

$1.$ Production Quantity for Maximum Profit: Determine the quantity of Product A and Product B that should be manufactured to achieve the highest profit $(0\mathrm{.}5$ points$)$

$2.$ Profit Projection: Determine the profit resulting from the optimal production mix $(0\mathrm{.}5$ points$)$

$3.$ Resource Utilization for Optimal Quantity: Provide details on the amount of material, machinery, and labor required for manufacturing the optimal quantity. Extract this information from the yellow column, which is the second last column in the Excel sheet $(0\mathrm{.}5$ points$)$

$4.$ Constraint Slack Analysis: Evaluate the surplus capacity or slack available for each constraint. Derive this information from the difference between the last column and the second last column $(0\mathrm{.}5$ points$)$

$5.$ Excel Calculation Snapshot: Present a high$-$quality snapshot of the Excel calculations, ensuring inclusion in this document $(0\mathrm{.}5$ points$)c.$ Maximize $Z=6A+3B(revenue)$

Subject $to$

Material $20A+6B600kg$

Machinery $25A+20B1,000hr$

Labour $20A+30B1,200hr$

$,A,B0$