Break-Even for a Weekend Market Stall
A vendor sells handmade candles at a weekend market. Fixed costs per market day are $140 for the stall fee plus $60 for transport, regardless of sales. Each candle costs $4.50 in materials and sells for $11. The vendor can make at most 90 candles per day. On a typical day she sells 45 candles, but sales vary with weather: on rainy days she sells about 60% of the typical figure. She is considering paying an extra $30 per day for a covered stall that she believes would remove the weather penalty.
1. Write the daily profit function and find the break-even quantity without the covered stall.
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Let n be candles sold. Revenue = 11n. Variable cost = 4.50n. Fixed cost = 140 + 60 = 200.
Profit P(n) = 11n − 4.50n − 200 = 6.50n − 200.
Break-even: 6.50n = 200, so n = 200/6.50 ≈ 30.8, meaning she must sell at least 31 candles to avoid a loss. Capacity is 90, so break-even is feasible. At the typical 45 candles, profit = 6.50(45) − 200 = 292.50 − 200 = $92.50.
2. Compare expected daily profit on a rainy day with and without the covered stall, and advise whether the upgrade is worth it on rainy days.
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Rainy sales without cover = 0.6 × 45 = 27 candles. Profit = 6.50(27) − 200 = 175.50 − 200 = −$24.50, a loss.
With cover, fixed cost rises to 230, but sales return to 45. Profit = 6.50(45) − 230 = 292.50 − 230 = $62.50.
The upgrade turns a $24.50 loss into a $62.50 profit, a swing of $87 per rainy day. It is clearly worth it whenever rain is expected. On a dry day, however, the cover costs $30 with no extra sales, reducing profit from $92.50 to $62.50, so the decision should be made day by day based on the forecast.
3. Suppose rain is forecast on 30% of market days. Compute the expected daily profit of always paying for the cover versus never paying, and identify the rain probability at which the two strategies are equal.
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Always cover: every day profit = $62.50, so expected = $62.50.
Never cover: dry day profit $92.50 (70% of days), rainy day profit −$24.50 (30%). Expected = 0.7(92.50) + 0.3(−24.50) = 64.75 − 7.35 = $57.40.
Always covering is better by $5.10 per day at a 30% rain rate.
Let p be the rain probability. Never-cover expected = (1−p)(92.50) + p(−24.50) = 92.50 − 117p. Set equal to 62.50: 92.50 − 117p = 62.50, so 117p = 30, giving p ≈ 0.256.
If rain is expected on more than about 26% of days, always paying for cover is the better fixed strategy; below that, paying only when rain is forecast is better.