Choosing a Printer: Linear Cost Functions and Break-Even
The school technology club is choosing a printing company for a fundraiser. Company A charges a $35 setup fee plus $0.12 per copy. Company B charges a $60 setup fee plus $0.08 per copy. The club expects to print between 500 and 900 copies, and it wants the lower total cost. A student writes 35 + 0.12c = 60 + 0.08c, finds c = 625, and concludes that Company B is cheaper because 625 is less than 900. Another student graphs both cost functions and says Company A is cheaper for every order under 625 copies. The advisor asks the group to justify which conclusion is valid over the actual order range, including what happens at 625 copies, and to explain what the slope and vertical intercept mean in this situation. The club also needs a recommendation if it can print exactly 520 copies, 625 copies, or 900 copies.
1. Write the cost functions A(c) and B(c), solve for the break-even number of copies, and state which company is cheaper at 520, 625, and 900 copies.
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Let c be the number of copies. Then A(c) = 35 + 0.12c and B(c) = 60 + 0.08c.
Set them equal: 35 + 0.12c = 60 + 0.08c. Subtract 0.08c: 35 + 0.04c = 60. Subtract 35: 0.04c = 25. Divide by 0.04: c = 625. At 625 copies the costs are equal.
At 520 copies: A = 35 + 0.12(520) = 35 + 62.40 = $97.40. B = 60 + 0.08(520) = 60 + 41.60 = $101.60. Company A is cheaper by $4.20.
At 625 copies: A = 35 + 75 = $110.00 and B = 60 + 50 = $110.00, so the costs are equal.
At 900 copies: A = 35 + 108 = $143.00 and B = 60 + 72 = $132.00. Company B is cheaper by $11.00.
2. Critique the first student's conclusion and the second student's statement. Which is mathematically valid over the club's range, and what qualification is needed?
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The first student found the correct break-even point but drew the wrong conclusion. The equation tells us that Company A is cheaper below 625 copies and Company B is cheaper above 625 copies. The fact that 625 is less than 900 does not mean Company B is cheaper for every order under 900; it is cheaper only after 625.
The second student's statement is essentially correct but incomplete. For 0 ≤ c < 625, Company A is cheaper because its lower setup fee outweighs its higher per-copy charge. At exactly 625 copies the costs are equal. For c > 625, Company B is cheaper because its lower per-copy charge has saved enough to recover the higher setup fee.
Over the club's expected range, the valid summary is: 500 to 624 copies favor Company A; 625 copies is a tie; 626 to 900 copies favor Company B, assuming whole-number copies.
3. Interpret the slope and y-intercept of each function in context. Then explain why the club cannot simply compare the slopes alone to choose a company for 520 copies.
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For Company A, the y-intercept 35 means the fixed setup fee is $35 even if zero copies are printed. The slope 0.12 means each additional copy adds $0.12 to the total cost. For Company B, the intercept 60 is the $60 setup fee, and the slope 0.08 means each additional copy adds $0.08.
Company B has the lower per-copy cost, but it also has the higher fixed cost. For a relatively small order such as 520 copies, the fixed cost matters a great deal. Company A starts $25 cheaper, and at 520 copies Company B's per-copy savings are only 520 × 0.04 = $20.80, which is not enough to overcome the $25 setup difference. Comparing slopes alone shows which company becomes cheaper as the order grows, but it does not show where that switch occurs. The intercept and the number of copies must both be used.