Question

select the correct answer.
elena makes banana bread and nut bread to sell at the market. a loaf of banana bread requires 2 cups of flour and 2 eggs. a loaf of nut bread takes 3 cups of flour and 1 egg. elena has 12 cups flour and 8 eggs on hand.
if she makes $1.50 profit per loaf of banana bread and $2 per loaf of nut bread, how many loaves of banana bread and nut bread could she make that will maximize her profit?
a. elena could make 1 loaf of banana bread and 5 loaves of nut bread to maximize her profit.
b. elena could make 0 loaves of banana bread and 4 loaves of nut bread to maximize her profit.
c. elena could make 2 loaves of banana bread and 3 loaves of nut bread to maximize her profit.
d. elena could make 3 loaves of banana bread and 2 loaves of nut bread to maximize her profit.

Explanation:

Step1: Define variables

Let $x$ = number of banana bread loaves, $y$ = number of nut bread loaves.

Step2: List constraints

Flour constraint: $2x + 3y \leq 12$
Egg constraint: $2x + y \leq 8$
Non-negativity: $x \geq 0, y \geq 0$

Step3: Define profit function

Profit $P = 1.5x + 2y$

Step4: Find feasible vertices

Solve intersection of constraints:

1. $x=0,y=0$: $P=0$
2. $x=0, 3y=12 \to y=4$: $P=1.5(0)+2(4)=8$
3. $y=0, 2x=8 \to x=4$: $P=1.5(4)+2(0)=6$
4. Solve

$$\begin{cases}2x+3y=12\\2x+y=8\end{cases}$$

Subtract equations: $2y=4 \to y=2$, substitute to get $2x+2=8 \to x=3$: $P=1.5(3)+2(2)=4.5+4=8.5$

1. Check option C: $x=2,y=3$: $2(2)+3(3)=4+9=13>12$ (violates flour constraint, invalid)
2. Check option A: $x=1,y=5$: $2(1)+3(5)=2+15=17>12$ (violates flour constraint, invalid)

Step5: Compare valid profits

Valid profits: $0, 8, 6, 8.5$

Answer:

D. Elena could make 3 loaves of banana bread and 2 loaves of nut bread to maximize her profit.