Question

1. 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?
3 batches of chocolate chip
2 batches of oatmeal raisin

1. you and your friend go to a store where all the shirts cost the same amount and all the pants cost same amount.

• you buy 1 shirt and 3 pairs of pants for $35.
• your friend buys 3 shirts and 2 pairs of pants for $42.

what is the cost for each shirt and each pair of pants?
each shirt costs $8
each pair of pants costs $9

Explanation:

For Question 18:

Step1: Define variables

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

Step2: List constraints

Eggs: $4x + 2y \leq 16$
Flour: $3x + 3y \leq 15$
Non-negativity: $x \geq 0, y \geq 0$

Step3: Simplify constraints

Eggs: $2x + y \leq 8$
Flour: $x + y \leq 5$

Step4: List feasible vertices

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

Step5: Calculate profit at vertices

Profit function: $P=4x+3y$

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

Step1: Define variables

Let $s$ = cost of 1 shirt, $p$ = cost of 1 pair of pants.

Step2: Set up equations

Your purchase: $s + 3p = 35$
Friend's purchase: $3s + 2p = 42$

Step3: Solve the system

Multiply first equation by 3: $3s + 9p = 105$
Subtract second equation: $(3s+9p)-(3s+2p)=105-42$
$\implies 7p=63 \implies p=9$

Step4: Find shirt cost

Substitute $p=9$ into $s+3p=35$:
$s + 3(9)=35 \implies s=35-27=8$

Answer:

3 batches of chocolate chip
2 batches of oatmeal raisin

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For Question 19: