Each machine is available $150$ hours per month, and the current size of the assembly department provides

capacity of $600$ hours. Each bookcase produced yields a unit profit contribution as follows:

Standard Narrow Wide

Profit $$8$ $$6$ $$10$

$(1)$ What is the optimal production quantity for $$Standard$$ the company? $(10$ points$)$

$(2)$ What is the shadow price on the Trimmer? $(5$ points$)$

$(3)$ What is the price you are willing to pay for one more hour of the Shaper? $(5$ points