The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering

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 730 homeless patients and no more than 900 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,100 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 $9,500 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

Explanation / Answer

Let the no of high income patients be i and the no of homeless patients be h

According to the conditions mentioned in the problem

0 <= i <= 900

h >= 730

i + h <= 2100

And h <= 2i

Profit = 9500h + 7600i

To maximize profit let us take the maximum value of i i.e 900

Therefore h becomes 1200 and these values satisfies all the conditions

Profit = 9500*1200 + 7600*900 = 18240000

Now, let's take a value such that h is maximazied

For that h = 2i

Therefore 3i =2100 and i = 700 and h =1400

Profit = 9500*1400 +7600*700 = 18620000

In the second case , the profit is maximized

Therefore answer

High income patients: 700

Homeless patients: 1400

Profit: 18620000

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