$1.$ Fancy Toy $(15$ points$)$

Fancy Toy Factory wants to determine their production mix, with the goal of maximizing profits. The factory produces toy cars and toy trucks. The profit on each car is $$3$ while the profit on each truck is $$5.$ The toys are manufactured with a machine called a plastic extruder which utilizes plastic resins. Each car requires $1$ lbs$.$ of plastic resins to produce while each truck require $2$ lbs$.$ The daily supply of plastic resins is limited to at most $1800$ pounds. Each car requires $2$ hours of machine time, and each truck also requires $2$ hours. At the moment the factory has a total of $2500$ hours available. In addition, at most a total of $2000$ toys can be sold.

Solver$$s sensitivity report, which was generated based on a linear model formulation for the problem, is provided below. Without solving the model explicitly $($i$.$e$.,$ by simply looking at the sensitivity report$),$ answer the following questions. Assume that the questions are independent $($e$.$g$.,$ Q$1\mathrm{.}2$ does not depend on Q$1\mathrm{.}1)$

Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$$4$ Car $7000320\mathrm{.}5$

$C$$4$ Truck $5500512$

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$D$$10$ Plastic Resins $180021800700550$

$D$$11$ Machine Hours $25000\mathrm{.}525001100700$

$D$$12$ Total Units $1252020001$E$+30748$

$1)$ Fancy Toys Inc. can procure additional $650$ pounds of resins at a cost of $$3$ per pound. Should it be purchased? Explain in $1$ sentence. $(5$ points$)$

$2)$ Suppose that Fancy Toys Inc. has an additional $$100$ to invest. It can allocate it to either renting out a machine at the cost of $$1$ per hour, or towards purchasing plastic resins at $$1$ per pound $($or both$).$ What should it do$?$ Explain in $1$ sentence. $(5$ points$)$

$3)$ How would the number of cars and trucks change if the profit margin on the cars increased by $$1?(5$ points$)$