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 \( x \) 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**: \( 4x + 2y \leq 16 \) (simplify: \( 2x + y \leq 8 \))
- **Flour Constraint**: \( 3x + 3y \leq 15 \) (simplify: \( x + y \leq 5 \))
- Non - negativity: \( x \geq 0, y \geq 0 \) (and \( x,y \) are integers)

Step3: Set Up Objective Function

The profit function \( P = 4x+3y \), which we want to maximize.

Step4: Find Feasible Region Vertices

1. Intersection of \( 2x + y=8 \) and \( x + y = 5 \):
Subtract the second equation from the first: \( (2x + y)-(x + y)=8 - 5\Rightarrow x = 3 \). Substitute \( x = 3 \) into \( x + y=5 \), we get \( y = 2 \).
2. Intersection of \( x + y=5 \) and \( y = 0 \): \( x=5,y = 0 \)
3. Intersection of \( 2x + y=8 \) and \( x = 0 \): \( y = 8 \), but from \( x + y\leq5 \), when \( x = 0 \), \( y\leq5 \), so the intersection of \( x = 0 \) and \( x + y=5 \) is \( (0,5) \)
4. Intersection of \( y = 0 \) and \( 2x + y=8 \): \( x = 4,y = 0 \)

Step5: Evaluate Profit at Vertices

- At \( (3,2) \): \( P=4\times3 + 3\times2=12 + 6=18 \)
- At \( (5,0) \): \( P=4\times5+3\times0 = 20 \)? Wait, no, check the flour constraint. If \( x = 5 \), flour used is \( 3\times5+3\times0=15 \), egg used is \( 4\times5+2\times0 = 20>16 \). So \( (5,0) \) is not in the feasible region.
- At \( (0,5) \): \( P=4\times0+3\times5 = 15 \)
- At \( (4,0) \): \( P=4\times4+3\times0=16 \)
- Wait, we made a mistake in the first vertex calculation. Let's re - check the constraints. The egg constraint is \( 4x + 2y\leq16 \), flour constraint \( 3x + 3y\leq15 \).
Let's list all integer points in the feasible region:
From \( x + y\leq5 \) and \( 2x + y\leq8 \) and \( x,y\geq0 \) integers.
Possible \( x \) values from 0 to 4 (since \( 4x\leq16\Rightarrow x\leq4 \))
- \( x = 0 \):
- \( y \) can be 0 - 5. \( P(0,y)=3y \). Max at \( y = 5 \), \( P = 15 \)
- \( x = 1 \):
- From \( x + y\leq5\Rightarrow y\leq4 \); from \( 2x + y\leq8\Rightarrow y\leq6 \). So \( y\leq4 \). \( P(1,y)=4 + 3y \). At \( y = 4 \), \( P=4 + 12 = 16 \)
- \( x = 2 \):
- From \( x + y\leq5\Rightarrow y\leq3 \); from \( 2x + y\leq8\Rightarrow y\leq4 \). So \( y\leq3 \). \( P(2,y)=8 + 3y \). At \( y = 3 \), \( P=8 + 9 = 17 \)
- \( x = 3 \):
- From \( x + y\leq5\Rightarrow y\leq2 \); from \( 2x + y\leq8\Rightarrow y\leq2 \). So \( y = 2 \). \( P(3,2)=12 + 6 = 18 \)
- \( x = 4 \):
- From \( x + y\leq5\Rightarrow y\leq1 \); from \( 2x + y\leq8\Rightarrow y\leq0 \). So \( y = 0 \). \( P(4,0)=16 \)

Wait, earlier mistake: when \( x = 5 \), egg usage is \( 4\times5=20>16 \), so \( x = 5 \) is not feasible. Let's re - express the egg constraint correctly. The egg constraint is \( 4x+2y\leq16 \), so when \( x = 3 \), \( 4\times3+2\times2=12 + 4 = 16 \) (correct), flour: \( 3\times3+3\times2=9 + 6 = 15 \) (correct). When \( x = 4 \), egg: \( 4\times4=16 \), flour: \( 3\times4=12\leq15 \). Wait, \( x = 4,y = 0 \): egg \( 16 \), flour \( 12 \). Profit \( P = 16 \). When \( x = 3,y = 2 \): profit \( 18 \). When \( x = 2,y = 3 \): egg \( 4\times2+2\times3=8 + 6 = 14\leq16 \), flour \( 3\times2+3\times3=6 + 9 = 15 \). Profit \( 4\times2+3\times3=8 + 9 = 17 \). When \( x = 1,y = 4 \): egg \( 4\times1+2\times4=4 + 8 = 12\leq16 \), flour \( 3\times1+3\times4=3 + 12 = 15 \). Profit \( 4\times1+3\times4=4 + 12 = 16 \). When \( x = 0,y = 5 \): egg \( 2\times5 = 10\leq16 \), flour \( 3\times5=15 \). Profit \( 15 \).

Wait, another way: Let's graph the inequalities.
The egg inequality \( 4x + 2y\leq16 \) or \( y\leq - 2x + 8 \)
The flour inequality \( 3x+3y\leq15 \) or \( y\leq - x + 5 \)

The feasible region is bounded by \( y\leq - x + 5 \), \( y\leq - 2x + 8 \), \( x\geq0 \), \( y\geq0 \)

The intersection of \( y=-x + 5 \) and \( y=-2x + 8 \) is at \( -x + 5=-2x + 8\Rightarrow x = 3 \), \( y = 2 \)

Now, let's check the profit function at all integer points in the feasible region:
- \( (3,2) \): \( 4*3 + 3*2=18 \)
- \( (4,0) \): \( 4*4+3*0 = 16 \)
- \( (2,3) \): \( 4*2+3*3=8 + 9 = 17 \)
- \( (1,4) \): \( 4*1+3*4=4 + 12 = 16 \)
- \( (0,5) \): \( 4*0+3*5 = 15 \)
- \( (3,2) \) gives a higher profit than \( (4,0) \) (18 vs 16). Wait, but let's check the egg and flour again for \( (3,2) \): eggs \( 4*3+2*2 = 16 \) (used all eggs), flour \( 3*3+3*2=15 \) (used all flour). For \( (4,0) \): eggs \( 16 \), flour \( 12 \). The profit at \( (3,2) \) is 18, at \( (4,0) \) is 16. At \( (2,3) \) is 17. So the maximum profit is at \( (3,2) \)

Answer:

3 batches of chocolate chip, 2 batches of oatmeal raisin