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 750 homeless patients and no more than 1,100 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 $9,900 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7,900 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

Since having homeless patients gives a profit of $9900 each every month while having high income patient gives a profit of $7900 each, we need to maximize the number of homeless patients to make maximum profit.

Now, let x be the number of high income patients. Since we cannot have more than double of homeless patients of the high income, let's have exactly double. So let 2x patients be homeless.

We also want, x<=1100 and 750<=2x (since max high income patients can be 1100 and min homeless patients must be 750)

Now, total capacity = 1800

So, x+ 2x= 1800

Or, x= 600 and 2x=1200

So required homeless patients = 1200

High income patients = 600

Profit: 1200*9900 + 600*7900 = $16,620,000