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

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

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

Answer

Answer:

4 batches of chocolate chip, 2 batches of oatmeal raisin

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

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

Step3: Rewrite constraints

Eggs: $y\leq\frac{16 - 3x}{2}$. Flour: $y\leq6 - x$.

Step4: Express profit function

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

Step5: Test corner points

Intersection of $y = 0$ and $3x+2y=16$ gives $x=\frac{16}{3}\approx5.33$ (not an integer). Intersection of $y = 0$ and $3x + 3y=18$ gives $x = 6$. Intersection of $x = 0$ and $3x+2y=16$ gives $y = 8$. Intersection of $x = 0$ and $3x+3y=18$ gives $y = 6$. Solve the system of equations $\begin{cases}3x + 2y=16\3x+3y=18\end{cases}$ Subtract the first equation from the second: $(3x + 3y)-(3x + 2y)=18 - 16$, so $y = 2$. Substitute $y = 2$ into $3x+2y=16$, we get $3x+2\times2=16$, $3x=12$, $x = 4$.

Step6: Evaluate profit at corner points

  • $(0,0)$: $P=4\times0+3\times0=0$.
  • $(6,0)$: $P=4\times6+3\times0 = 24$.
  • $(0,6)$: $P=4\times0+3\times6=18$.
  • $(4,2)$: $P=4\times4+3\times2=16 + 6=22$.

The maximum profit occurs when $x = 4$ and $y = 2$.