Math 1 · Unit 6 · Lesson 6 · part 3 of 4
Two limits, two inequalities
The other kind of story has two limits instead of two facts: a budget and a time limit, a weight limit and...
Two limits, two inequalities
The other kind of story has two limits instead of two facts: a budget and a time limit, a weight limit and a space limit. Each limit is a constraint, an inequality (Unit 5, Lesson 4), and the two together are a system of inequalities. The feasible region, the overlap, is every plan that obeys both limits at once.
The story
You make bracelets (x) and keychains (y) for a fair. Materials cost $2 per bracelet and $1 per keychain, and you have at most $20. Each bracelet takes 30 minutes and each keychain 15 minutes, and you have at most 6 hours, which is 360 minutes.
The two constraints
Money: 2x + y ≤ 20. Time: 30x + 15y ≤ 360, which simplifies (multiply by 115) to 2x + y ≤ 24. And x ≥ 0, y ≥ 0.
Wait: both boundaries have the same slope. The money line is lower, so the money limit is the tighter one, and every plan under $20 is automatically under 6 hours. The overlap is just the money region.
A second story, with a real overlap
Same bracelets and keychains, but now bracelets take 45 minutes: 45x + 15y ≤ 360, which is 3x + y ≤ 24. Now the two limits cut across each other, and the overlap is a four-sided region in the corner. A plan like 5 bracelets and 8 keychains: money 18 ≤ 20, time 23 ≤ 24. Feasible.