A linear cost model adds a one-time fee to a charge for each item. When a question asks when one shop becomes cheaper, compare the totals with a strict inequality. Find where the costs tie using a Desmos regression, then use the per-part rates to decide which side of that cutoff is cheaper. Parts must be whole, so don't round the cutoff to the nearest integer.
Hints
- Hint 1
A set-up fee is paid once, even if the shop makes many parts. If counts parts, add that fee once and multiply each per-part price by . What inequality says the first shop is cheaper?
- Hint 2
The break-even point is where both shops charge the same amount. Set their cost expressions equal in Desmos to find that cutoff. A tie does not satisfy a request for one cost to be less than the other.
- Hint 3
The first shop charges less for each added part, so it becomes cheaper as the count rises past the tie. Because parts come in whole numbers, which direction should you move from the cutoff to find the minimum?
Step-by-step
Find the equal-cost cutoff
Step 1Compare the two total costs
Let be the number of parts in one production run. The first shop adds the one-time set-up fee of $40 to its $2.35 charge for each part. The competing shop charges $3.10 for each part and has no fee. So the first shop's cost must be less than the competing shop's cost: .
- Step 2
Find where the costs tie
The break-even point is where the two costs are equal, so use the same cost expressions to find that boundary. Type in Desmos. The asks Desmos to find a value that makes both sides equal, and tells it to solve for the part count rather than graph . Under PARAMETERS, Desmos shows . That's a tie, not a count where the first shop is cheaper.
- Step 3
Choose the first whole count that works
Past the tie, the first shop adds only $2.35 for each new part while the competitor adds $3.10, so the first shop is cheaper above the cutoff. For a strict inequality with whole parts, move past the tie to the first integer on the cheaper side. That is 54 parts, the minimum that works. Grid in 54.