Complete the following linear programming problem Show all of your steps. Including the complete systems of inequalities, declaring the variables and their units, your Excel work, and an interpretation of your solution.

Your dorm roommate has an idea for a mirco$-$business with only up to $$100$ of funds per month. They are trying to learn using the Makerspace at VCU's library while also making some money. They plan to make assorted $3$D prints as well as engrave cute pet collar tags on wood and metal.$$ The $3$D prints vary from small $50$g prints that take about $3$ hours to print, medium $75$g prints that take closer to $6$hrs$,$ and large $125$g prints that take $9$ hours. They sell on Etsy the $3$D prints that are small for $$9\mathrm{.}50,$ medium for $$13\mathrm{.}50,$ and large for $$22\mathrm{.}50.$ The metal and wooden pet tags are sold for $$8$ and $$10$ respectively. They usually use ABS filament that costs $$0\mathrm{.}10/$g for all of their $3$D prints.$$

They only have about $20$ hours available to them in a month to print and create these items. Assuming that printing and engraving doesn't cost any time, as they can study in the library while printing, they spend on average $1$ hour gathering wood supplies from parks nearby and carving them into shape for each dog tag that is made of wood $($they want this hour of work to be a minimum wage cost as they cannot study while doing it$).$ The metal dog tags cost $$5$ per tag for a blank version and they engrave them in $1$hr $($again they can study through this process$).$

Maximize monthly profit knowing that Etsy gets a $6\mathrm{.}5\%$ of all listed prices.

Fully analyze the results that you get. Do they make sense given the situation? Explain why in hindsight that is the optimal solution given the context of the problem.