Help me with this word problem?

Question

A manufacturer of calculators produces two models: standard and scientific. Long-term demand for the two models mandates that the company manufacture at least 110 standard and 40 scientific calculators each day. However, because of limitations on production capacity, no more than 200 standard and 160 scientific calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped every day.

(a) If the production cost is $5 for a standard calculator and $7 for a scientific one, how many of each model should be produced daily to minimize this cost?

(b) If each standard calculator results in a $2 loss but each scientific one produces a $5 profit, how many of each model should be made daily to maximize profit?

Answer 1

Graph the constraints; I chose x = # of standard and y = # of scientific calculators. This gives a five-sided area that satisfies x ≥ 110, x ≤ 200, y ≥ 40, y ≤ 160, and x + y ≥ 200. The vertices of this figure are (110,90), (110,160), (200,160), (200,40), and (160,40).

At each vertex calculate (a) 5x + 7y and (b) -2x + 5y. For (a) this yields 160 standard and 40 scientific to minimize cost. For (b) it is 110 standard and 160 scientific which maximizes profit.