The number of Medicarecertified home-health-care agencies (...

The number of Medicarecertified home-health-care agencies ( $70 \%$ are freestanding, and $30 \%$ are owned by a hospital or other large facility) has been declining at the rate of $$ 0.186 e^{-0.02 t} \quad(0 \leq t \leq 14) $$ thousand agencies/year between $1988(t=0)$ and 2002 $(t=14)$. The number of such agencies stood at $9.3$ thousand units in 1988 . a. Find an expression giving the number of health-care agencies in year $t$. b. What was the number of health-care agencies in $2002 ?$ c. If this model held true through 2005 , how many care agencies were there in 2005 ?

The number of Medicarecertified home-health-care agencies ( $70 \%$ are freestanding, and $30 \%$ are owned by a hospital or other large facility) has been declining at the rate of
$$
0.186 e^{-0.02 t} \quad(0 \leq t \leq 14)
$$
thousand agencies/year between $1988(t=0)$ and 2002 $(t=14)$. The number of such agencies stood at $9.3$ thousand units in 1988 .
a. Find an expression giving the number of health-care agencies in year $t$.
b. What was the number of health-care agencies in $2002 ?$
c. If this model held true through 2005 , how many care agencies were there in 2005 ?

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Key Concepts

Integration of Rate Functions

This concept involves summing a continuously changing rate over time to determine the total change. When the rate of change of a quantity is provided, integration is used to accumulate these incremental changes and thereby obtain a function that represents the total quantity at any time.

Initial Value Problems

In many situations, a differential equation must be solved in conjunction with an initial condition to uniquely determine the solution. This process involves finding the integration constant after integrating the rate function, ensuring that the solution accurately reflects the starting value provided by the problem.

Exponential Decay

Exponential decay describes processes where the rate at which a quantity decreases is proportional to its current magnitude, leading to a rapid initial drop that slows over time. Recognizing and working with exponential functions is essential when dealing with models that incorporate decreasing rates such as the one in this problem.

Model Extrapolation

Extrapolation involves extending a mathematical model beyond the range for which data was originally observed. It is important to understand the assumptions and limitations of the model when making predictions in a time period that was not initially covered, ensuring the reliability of such forecasts.

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