a baker has 16 eggs and 18 cups of flour.\n• one batch of chocolate chip cookies requires 3 eggs and 3 cups…

a baker has 16 eggs and 18 cups of flour.\n• one batch of chocolate chip cookies requires 3 eggs and 3 cups of flour.\n• one batch of oatmeal raisin cookies requires 2 eggs and 3 cups of flour.\nthe baker makes $4 profit for each batch of chocolate chip cookies and $3 profit for each batch of oatmeal raisin cookies.\nhow many batches of each type of cookie should she make to maximize profit?\nbatches of chocolate chip\nbatches of oatmeal raisin

a baker has 16 eggs and 18 cups of flour.\n• one batch of chocolate chip cookies requires 3 eggs and 3 cups of flour.\n• one batch of oatmeal raisin cookies requires 2 eggs and 3 cups of flour.\nthe baker makes $4 profit for each batch of chocolate chip cookies and $3 profit for each batch of oatmeal raisin cookies.\nhow many batches of each type of cookie should she make to maximize profit?\nbatches of chocolate chip\nbatches of oatmeal raisin

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of batches of chocolate - chip cookies and $y$ be the number of batches of oatmeal - raisin cookies.

Step2: Set up constraints

Egg constraint: $3x + 2y\leq16$. Flour constraint: $3x+3y\leq18$. Also, $x\geq0,y\geq0$ and $x,y$ are non - negative integers.

Step3: Rewrite constraints

From $3x + 3y\leq18$, we get $y\leq6 - x$. From $3x+2y\leq16$, we get $y\leq\frac{16 - 3x}{2}$.

Step4: Express the profit function

The profit function $P = 4x+3y$.

Step5: Find the feasible region

We can test the corner points of the feasible region. If $x = 0$: From $3x+3y = 18$, $y = 6$ and $P=3\times6 = 18$. From $3x + 2y=16$, $y = 8$ (but $3x+3y>18$ when $y = 8$ for $x = 0$, so we consider the intersection with $3x + 3y=18$). If $y = 0$: From $3x+3y = 18$, $x = 6$ and $P=4\times6=24$. From $3x + 2y=16$, $x=\frac{16}{3}\approx5.33$ (we take the integer value $x = 5$), and $P = 4\times5=20$. Now find the intersection of $3x + 3y=18$ (i.e., $y = 6 - x$) and $3x+2y=16$. Substitute $y = 6 - x$ into $3x+2y=16$: $3x+2(6 - x)=16$. $3x + 12-2x=16$. $x=4$. When $x = 4$, $y=6 - 4=2$. $P=4\times4 + 3\times2=16 + 6=22$.

Answer:

4 batches of chocolate chip 2 batches of oatmeal raisin