A fixed total of two kinds of items and a time cap lead to a linear inequality. Name the number the question asks for, then write the other count using the fixed total. Each time you replace a faster item with a slower one, the time used increases, so the maximum occurs at the time limit. Use a Desmos regression to find that boundary; dividing all the available time by the slower item's time would forget the other items.
Hints
- Hint 1
Let be the number of necklaces. Because there are exactly pieces, the bracelet count is , not another independent number. How can you use both per-piece times to write the total time?
- Hint 2
An upper bound is a largest allowed value. Replacing one bracelet with one necklace uses more time. How many extra minutes does each swap take, and what happens when the jeweler reaches the time limit?
Step-by-step
Model the time and find its limit
Step 1Write the time limit
Let be the number of necklaces. With exactly pieces, there are bracelets. Bracelets take minutes each and necklaces take minutes each, so the inequality for using at most minutes is .
- Step 2
See how each necklace changes the time
If all pieces were bracelets, they would take minutes. Replacing a bracelet with a necklace adds minutes, so you can rewrite the limit as . Each added necklace raises the time used. To maximize the necklace count, find where the time reaches the cap.
- Step 3
Find and check the maximum
For the time boundary, type in Desmos. The tilde asks Desmos to find the value of that uses exactly minutes; it shows under PARAMETERS. That leaves bracelets. One more necklace would add minutes and exceed the cap, so the maximum is necklaces. Choice C.