Question

a home improvement store offers two different types of decorative lights. a small light costs the store $17 to purchase and a large light costs $28 each. in order to get free shipping, the store must purchase at least $3,000 worth of lights. the truck that the lights are shipped in can haul a maximum of 150 boxes of lights. which of the following combinations of small and large lights would not be a possible combination for this scenario? 80 small lights and 60 large lights 90 small lights and 65 large lights 55 small lights and 80 large lights 75 small lights and 70 large lights

Explanation:

Step1: Define Variables

Let \( x \) be the number of small lights (cost \( \$17 \) each) and \( y \) be the number of large lights (cost \( \$28 \) each). We have two constraints:

1. \( x + y \leq 150 \) (max boxes)
2. \( 17x + 28y \geq 3000 \) (min cost for free shipping)

Step2: Check Option A (80, 60)

• Total boxes: \( 80 + 60 = 140 \leq 150 \)
• Total cost: \( 17(80) + 28(60) = 1360 + 1680 = 3040 \geq 3000 \) → Valid

Step3: Check Option B (90, 65)

• Total boxes: \( 90 + 65 = 155 > 150 \) → Violates box constraint. Wait, wait, miscalculation? Wait \( 90 + 65 = 155 \), which is more than 150. But wait, let's recalculate. Wait, maybe I misread. Wait the options:

Wait the options are:

A. 80 small, 60 large

B. 90 small, 65 large

C. 55 small, 80 large

D. 75 small, 70 large

Wait let's recheck each:

Option A: \( x=80, y=60 \)

• Boxes: \( 80 + 60 = 140 \leq 150 \)
• Cost: \( 17*80 + 28*60 = 1360 + 1680 = 3040 \geq 3000 \) → Valid

Option B: \( x=90, y=65 \)

• Boxes: \( 90 + 65 = 155 > 150 \) → Invalid (exceeds box limit)

Option C: \( x=55, y=80 \)

• Boxes: \( 55 + 80 = 135 \leq 150 \)
• Cost: \( 17*55 + 28*80 = 935 + 2240 = 3175 \geq 3000 \) → Valid

Option D: \( x=75, y=70 \)

• Boxes: \( 75 + 70 = 145 \leq 150 \)
• Cost: \( 17*75 + 28*70 = 1275 + 1960 = 3235 \geq 3000 \) → Valid

Wait but the question is "which combination would NOT be possible". So Option B (90,65) has \( 90 + 65 = 155 > 150 \), so it's invalid. But wait, maybe I made a mistake. Wait the problem says "the truck can haul a maximum of 150 boxes". So total boxes (small + large) must be ≤150. So for B: 90 + 65 = 155 > 150 → not possible.

Wait but let's confirm the cost too, but the box constraint is violated first. So B is invalid.

Wait but let's check again. Wait the options:

Wait the user's image:

Options:

1. 80 small, 60 large

1. 90 small, 65 large

1. 55 small, 80 large

1. 75 small, 70 large

So Step2: Check B (90,65):

Total boxes: 90 + 65 = 155 > 150 → violates the max box constraint. So this combination is not possible.

Answer:

B. 90 small lights and 65 large lights