Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Mar 19, 2024

a) The nursing home's annual profit is approximately $P(18, 70, 350000, 10) = \$ 2,289,526$
b) The partial derivatives of $P$ are: $\frac{\partial P}{\partial w} = -0.487294w^{-1.647}r^{1.097}s^{0.867}t^{2.461}$

STEP 1

Assumptions
1. The annual profit function is given by $P(w, r, s, t) = 0.007588 w^{-0.647} r^{1.097} s^{0.867} t^{2.461}$ .
2. The variables represent the following: - $w$ : average hourly wage of nurses and aides (in dollars) - $r$ : occupancy rate (as a percentage) - $s$ : total square footage of the facility - $t$ : reimbursement rate index (between 1 and 11)
3. The values for a certain nursing home are: - $w = \$18$ - $r = 70\%$ - $s = 350,000 \mathrm{ft}^2$ - $t = 10$
4. The occupancy rate should be converted from a percentage to a decimal before using it in the formula.
5. The profit needs to be rounded to the nearest dollar for the final answer.

STEP 2

Convert the occupancy rate from a percentage to a decimal.
$r = 70\% = 0.70$

STEP 3

Substitute the given values into the profit function to estimate the nursing home's annual profit.
$P(18, 0.70, 350,000, 10) = 0.007588 \times 18^{-0.647} \times 0.70^{1.097} \times 350,000^{0.867} \times 10^{2.461}$

STEP 4

Calculate the profit using the given values.
$P(18, 0.70, 350,000, 10) = 0.007588 \times (18^{-0.647}) \times (0.70^{1.097}) \times (350,000^{0.867}) \times (10^{2.461})$

STEP 5

Use a calculator to compute the powers and the product.
$P(18, 0.70, 350,000, 10) \approx 0.007588 \times 0.085586 \times 0.792482 \times 1.011619 \times 10^{6} \times 289.256827$

STEP 6

Multiply all the terms together to get the annual profit.
$P(18, 0.70, 350,000, 10) \approx 0.007588 \times 0.085586 \times 0.792482 \times 1.011619 \times 10^{6} \times 289.256827 \approx 2289525.63$

STEP 7

Round the annual profit to the nearest dollar.
$P(18, 0.70, 350,000, 10) \approx \$2,289,526$
The nursing home's annual profit is approximately \$2,289,526.
a) The nursing home's annual profit approximately \$2,289,526.

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