Question

select the correct answer.

ms. waller receives orders for posy bouquets and cascade bouquets. it usually takes her 15 minutes to make one posy bouquet and 20 minutes to make one cascade bouquet. for todays orders, each posy bouquet must have 12 stems and each cascade bouquet must have 6 stems. ms. waller has no more than 300 minutes to make the bouquets, and she can procure no more than 180 stems in all.
ms. waller makes a profit of $10 on each posy bouquet and $7 on each cascade bouquet. how many posy bouquets and cascade bouquets should she deliver to maximize her profit?

a. ms. waller should make 12 posy bouquets and 6 cascade bouquets to maximize her profit.

b. ms. waller should make 6 posy bouquets and 12 cascade bouquets to maximize her profit.

c. ms. waller should make 15 posy bouquets and 0 cascade bouquets to maximize her profit.

d. ms. waller should make 0 posy bouquets and 15 cascade bouquets to maximize her profit.

Explanation:

Step1: Define Variables

Let \( x \) be the number of posy bouquets and \( y \) be the number of cascade bouquets.

Step2: Set Up Constraints

• Time constraint: \( 15x + 20y \leq 300 \) (simplify by dividing by 5: \( 3x + 4y \leq 60 \))
• Stem constraint: \( 12x + 6y \leq 180 \) (simplify by dividing by 6: \( 2x + y \leq 30 \))
• Non - negativity: \( x\geq0,y\geq0 \)

Step3: Objective Function

The profit function \( P \) to maximize is \( P = 10x + 7y \)

Step4: Find Feasible Region Vertices

1. Intersection of \( 3x + 4y=60 \) and \( 2x + y = 30 \)

• From \( 2x + y=30 \), we get \( y = 30 - 2x \)
• Substitute into \( 3x + 4y=60 \): \( 3x+4(30 - 2x)=60 \)
• \( 3x + 120-8x = 60 \)
• \( - 5x=60 - 120=-60 \)
• \( x = 12 \)
• Then \( y=30 - 2\times12 = 6 \)

1. Intersection of \( 3x + 4y = 60 \) and \( y = 0 \):

• \( 3x=60\Rightarrow x = 20 \), but check stem constraint: \( 2x+y=40>30 \), so not feasible.

1. Intersection of \( 2x + y=30 \) and \( x = 0 \):

• \( y = 30 \), check time constraint: \( 3\times0+4\times30 = 120\leq60 \)? No, not feasible.

1. Intersection of \( x = 0 \) and \( y = 0 \): \( (0,0) \)
2. Intersection of \( y = 0 \) and \( 2x + y=30 \): \( x = 15,y = 0 \) (check time constraint: \( 3\times15+4\times0=45\leq60 \))

Step5: Evaluate Profit at Vertices

• At \( (0,0) \): \( P=10\times0 + 7\times0=0 \)
• At \( (15,0) \): \( P = 10\times15+7\times0 = 150 \)
• At \( (12,6) \): \( P=10\times12 + 7\times6=120 + 42=162 \)
• At \( (0,15) \) (from \( 3x + 4y=60,y = 15,x = 0 \)): Check stem constraint \( 2\times0+15 = 15\leq30 \), time constraint \( 3\times0 + 4\times15=60\leq60 \), \( P=10\times0+7\times15 = 105 \)

Answer:

A. Ms. Waller should make 12 posy bouquets and 6 cascade bouquets to maximize her profit.