Break-even with a piecewise cost
A student society prints T-shirts to fund an event. The printer charges a fixed setup fee of 180 euros plus 4.20 euros per shirt for the first 100 shirts. Beyond 100 shirts the per-unit price drops to 3.60 euros for every additional shirt, but the setup fee is unchanged. The society sells shirts at 12 euros each but expects to give away 8 shirts to volunteers and to have 5 percent of the remaining printed shirts unsold (fractional shirts are possible in the model). Let x be the number of shirts printed, with x >= 8. Revenue comes only from sold shirts. The treasurer writes profit as a single expression and then discovers that the formula behaves differently on either side of x = 100.
1. Write the profit function P(x) as a piecewise expression for 8 <= x <= 100 and for x > 100, and evaluate P at x = 100 and x = 120.
Read the worked answer
Sold shirts = 0.95(x - 8). Revenue R(x) = 12 * 0.95(x - 8) = 11.4(x - 8).
Cost for x <= 100: C(x) = 180 + 4.2x. Cost for x > 100: C(x) = 180 + 420 + 3.6(x - 100) = 600 + 3.6(x - 100) = 240 + 3.6x.
So P(x) = 11.4(x - 8) - (180 + 4.2x) for 8 <= x <= 100, and P(x) = 11.4(x - 8) - (240 + 3.6x) for x > 100.
At x = 100: 11.4(92) - (180 + 420) = 1048.8 - 600 = 448.8. At x = 120: 11.4(112) - (240 + 432) = 1276.8 - 672 = 604.8.
Check continuity at 100 from the right: 240 + 360 = 600, matching, so cost is continuous; profit is therefore continuous at 100.
2. Find the smallest whole number of shirts that makes profit nonnegative in each region, and state the overall break-even print run.
Read the worked answer
Region 1: 11.4x - 91.2 - 180 - 4.2x >= 0 gives 7.2x >= 271.2, so x >= 37.67, i.e. x = 38 (and 38 <= 100).
Region 2: 11.4x - 91.2 - 240 - 3.6x >= 0 gives 7.8x >= 331.2, so x >= 42.46, i.e. x = 43.
Since 38 is in region 1 and is smaller, the overall break-even is 38 shirts. The region-2 threshold is irrelevant because profit is already positive before x = 100 and stays positive after (slope 7.8 > 0).
The lesson: solve each piece separately, then check that each solution lies in its own interval before comparing.
3. The treasurer claims that because the per-unit price falls after 100, the marginal profit must jump up at x = 100. Evaluate the marginal profit on each side and judge the claim.
Read the worked answer
Marginal profit is the slope. Region 1 slope: 11.4 - 4.2 = 7.2 euros per shirt. Region 2 slope: 11.4 - 3.6 = 7.8 euros per shirt.
So the slope does increase at x = 100, from 7.2 to 7.8, confirming the direction of the treasurer's claim. But there is no jump in profit itself: P is continuous at 100 (both pieces give 448.8), only the derivative jumps.
Precisely, the claim should be restated: the marginal profit rises by 0.6 euros per shirt at the 100-shirt threshold, while total profit is continuous there. A jump in P would require a discontinuity in cost, which does not occur because the discount applies only to units beyond 100.