EcoCharge, a company that produces solar-powered phone chargers, is planning to open a new assembly facility and is evaluating three potential locations. Sunvale has fixed costs of $21,000 per year and variable costs of $5 per charger, Brighton has fixed costs of $30,000 per year and variable costs of $3 per charger, and Lumina has fixed costs of $42,000 per year and variable costs of $2 per charger. If the expected annual demand for chargers is uncertain, in which range of annual demand would each location be the most cost-effective choice?

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We start by comparing the total annual cost for each location as a function of annual quantity x (units): - Sunvale: Cost = 21,000 + 5x - Brighton: Cost = 30,000 + 3x - Lumina: Cost = 42,000 + 2x Next, we find the break-even points where pairs of locations have equal cost to determine ranges where one is cheaper. - Sunvale vs Brighton: 21,000 + 5x = 30,000 + 3x → 2x = 9,000 → x = 4,500. For x < 4,500, Sunvale is cheaper; for x > 4,500, Brighton is cheaper. - Sunvale vs Lumina: 21,000 + 5x = 42,000 + 2x → 3x = 21,000 → x = 7,000. For x < 7,000, Sunvale is cheaper; for x > 7,000, Lumina is cheaper. - Brighton vs Lumina: 30,000 + 3x = 42,000 + 2x → x = 12,000. For x < 12,000, Brighton is cheaper; for x > 12,000, Lumina is cheaper. Now we combine these pairwise results to map out the cheapest option across the demand ranges: - From x = 0 up to x = 4,500: Sunvale is cheaper than Brighton (and also cheaper than Lumina, since Lumina only becomes cheaper than Sunvale after x = 7,000). Thus Sunvale is the cheapest in this band. - From just above x = 4,500 up to x = 12,000: Brighton is cheaper than Sunvale (due to the 2x gap after 4,500) and still cheaper than Lumina up to x = 12,000. Hence Brighton is the cheapest in this interval. - From x = 12,000 upward: Lumina becomes the cheapest, since Lumina undercuts Brighton after 12,000 and also undercuts Sunvale after 7,000, and continues to be best beyond 12,000. This yields the following clean interval assignments: - 0 to 4,500 units: Sunvale - 4,501 to 12,000 units: Brighton - 12,001+ units: Lumina Evaluating the options against these break-even analyses, the option that matches these interval boundaries is: Brighton: 0–4,500 units; Sunvale: 4,501–12,000 units; Lumina: 12,001 + units. The other options either swap the order of Sunvale and Brighton around the 4,500-unit mark or place Lumina in a different range, which contradicts the calculated cost comparisons above.

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