Crumbles Bakery needs to decide how many and what types of cupcakes to make today. Currently, they make two types of cupcakes: chocolate cupcakes and carrot cake cupcakes. Chocolate cupcakes sell for a profit of $1 per cupcake, while carrot cake cupcakes sell for a much higher profit of $4 per cupcake. There are two main ingredients in making their cupcakes: flour and sugar. Each chocolate cupcake requires 4 cups of flour and 2 cups of sugar. Each carrot cake cupcake requires 3 cups of flour and 4 cups of sugar. Crumbles has a total of 40 cups of flour and 36 cups of sugar to use. Also, they would like to make at least 3 carrot cake cupcakes. In order to determine how many of each type of cupcake to make, Crumbles has setup the linear program below. Find the optimal number of each type of cupcake to make and the optimal profit of Crumbles by solving the linear program. Use both (a) the enumerating the corner points method and (b) the iso-profit line method. Also identify the binding and non-binding constraints and the slack of each constraint.
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maximize X_{l }+ 4X_{2}
subject to
4X_{I} + 3X_{2} < 40
2X_{I} + 4X_{2} < 36
X_{2} > 3
X_{l}, X_{2} > 0