Math, microeconomics """help please

Question

The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 740 homeless patients and no more than 800 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 1,800 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $9,600 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7,600 per month. How many of each type of patient should it house in order to maximize profit? HINT [See Example 3.] (If an answer does not exist, enter DNE.)

high-income patients .......?

homeless patients ........?

profit $ ....?

Answer 1

homeless = x> 740

high income = y< 800

x+y = 1800

x<2y

2y+y = 1800

y =1800/3 = 600 high income < 800

x=2y =1200 homeless > 740

Profit = P = 9600x +7600y = 9600(1200)+7600(600)

= 11,520,000+4,560,000= 16,080,000 dollars

any increase in high income patients above 600 will be at the expense of an equal number of homeless patients and reduce profts