T-Shirt Fundraiser Order Size
The North Ridge robotics club is selling T-shirts to raise money for a competition. The supplier charges $8.50 per shirt plus a $120 setup fee. The club plans to sell each shirt for $15. Their goal is a profit of at least $600. A student writes the inequality 15x - 8.50x >= 600, solves it to get x >= 92.3, and says, 'We should order 92 shirts because 92 is the closest whole number.' The supplier then offers a bulk option: if the club orders at least 100 shirts, the price drops to $7.75 per shirt and the setup fee is waived, but unsold shirts cannot be returned. Last year the club sold 85 shirts, and members expect similar demand. The club adviser asks the students to correct the inequality, compare the regular and bulk offers, and recommend an order size that balances the $600 goal against the risk of unsold shirts.
1. Identify the student's error, solve the corrected inequality, and explain how the context determines the rounding.
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The student left out the $120 setup fee. Correct profit is 15x - 8.50x - 120, so the inequality is 15x - 8.50x - 120 >= 600. This simplifies to 6.50x - 120 >= 600, then 6.50x >= 720, so x >= 110.77. Because the club cannot order a fraction of a shirt and must meet at least $600, it must round up to 111 shirts. Rounding down to 92 is not justified by 'closest whole number'; the direction of rounding is set by the inequality. Check: 110 shirts give 6.50(110) - 120 = 595, below $600; 111 shirts give 6.50(111) - 120 = 601.50, which meets the goal. The student also omitted the fixed cost, which is the larger error.
2. Compare the profit if the club orders 111 shirts at the regular price and sells all of them with the profit if it orders 100 shirts at the bulk price and sells all of them. Which order meets the $600 goal?
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Regular price, 111 shirts: revenue is 111(15) = $1,665. Cost is 111(8.50) + 120 = $943.50 + $120 = $1,063.50. Profit is $1,665 - $1,063.50 = $601.50, so it barely meets the goal. Bulk price, 100 shirts: revenue is 100(15) = $1,500. Cost is 100(7.75) = $775 because the setup fee is waived. Profit is $1,500 - $775 = $725, which meets the goal with $125 to spare. Both meet the $600 goal if all shirts sell, but the bulk order yields $123.50 more profit and costs less up front. The bulk order is financially better, provided the club can sell enough shirts to avoid being stuck with too much unsold inventory.
3. Suppose the club orders 100 shirts at the bulk price but sells only 85, as last year. Calculate the profit and the number of shirts that must be sold to reach the $600 goal. What does this imply about the recommendation?
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With 85 sales at $15, revenue is 85(15) = $1,275. The cost is still 100(7.75) = $775, so profit is $500. To reach $600, solve 15n - 775 >= 600. Then 15n >= 1,375, so n >= 91.67, which means 92 shirts must be sold. At 85 sales the club misses the goal by $100. For comparison, ordering 111 at regular price and selling 85 gives revenue $1,275 and cost $1,063.50, for a profit of only $211.50. The bulk order is still much better at 85 sales. The recommendation should be to take the bulk order only if the club can reasonably pre-sell at least 92 shirts; otherwise it should reduce the order or find a pre-order system, because the $600 goal is not guaranteed by the lower price alone.