Question

a baker has 16 eggs and 15 cups of flour. one batch of chocolate chip cookies requires 4 eggs and 3 cups of flour. one batch of oatmeal raisin cookies requires 2 eggs and 3 cups of flour. the baker makes $4 profit for each batch of chocolate chip cookies and $3 profit for each batch of oatmeal raisin cookies. how many batches of each type of cookie should she make to maximize profit? batches of chocolate chip batches of oatmeal raisin

Explanation:

Step1: Define variables

Let $x$ = batches of chocolate chip, $y$ = batches of oatmeal raisin.

Step2: List constraints

Egg constraint: $4x + 2y \leq 16$ (simplifies to $2x + y \leq 8$)
Flour constraint: $3x + 3y \leq 15$ (simplifies to $x + y \leq 5$)
Non-negativity: $x \geq 0$, $y \geq 0$

Step3: Define profit function

Profit $P = 4x + 3y$

Step4: Find feasible vertices

Solve intersection of $2x + y = 8$ and $x + y = 5$:
Subtract: $(2x + y) - (x + y) = 8 - 5$ → $x = 3$
Substitute $x=3$ into $x + y =5$: $y=2$
Feasible vertices: $(0,0)$, $(0,5)$, $(3,2)$, $(4,0)$

Step5: Calculate profit at vertices

• $(0,0)$: $P = 4(0)+3(0) = 0$
• $(0,5)$: $P = 4(0)+3(5) = 15$
• $(3,2)$: $P = 4(3)+3(2) = 18$
• $(4,0)$: $P = 4(4)+3(0) = 16$

Answer:

3 batches of chocolate chip
2 batches of oatmeal raisin