provide solution for

$($a$)$ Suppose that instead of $$100,$ each sapphire costs $$120.$ Would Zales still purchase sapphires? If yes,what would be the new optimal profit? If no$,$ write NO$.$

Answer: $

$($b$)$ What is the most that Zales would be willing to pay for another sapphire on top of the regular price? $($i$.$e$.,$ what's ruby's shadow price?$)$

Answer: $

$($c$)$ What is the most that Zales would be willing to pay for another ruby?

Answer: $

$($d$)$ Suppose that Zales were only required to produce at least $23$ Type $2$ rings. What would Zales' profit now be$?$

Answer: $

$($e$)$ Zales is considering selling a new jewelry set consisting of $4$ sapphires and $2$ rubies. What should be the minimum selling price on this set for this to make sense?

Answer: