The exponential models describe the population of the...

The exponential models describe the population of the indicated country, A, in millions, t years after 2006. $$\begin{array}{|l|l|} \hline \text { India } & A=1095.4 e^{0.014} \\ \hline \text { Ina } & A=26.8 e^{0.027} \\ \hline \text { Japan } & A=127.5 e^{0.001 t} \\ \hline \text { Russia } & A=142.9 e^{-0.004 r} \\ \hline \end{array}$$ Which country has a decreasing population? By what percentage is the population of that country decreasing each year?

The exponential models describe the population of the indicated country, A, in millions, t years after 2006.
$$\begin{array}{|l|l|}
\hline \text { India } & A=1095.4 e^{0.014} \\
\hline \text { Ina } & A=26.8 e^{0.027} \\
\hline \text { Japan } & A=127.5 e^{0.001 t} \\
\hline \text { Russia } & A=142.9 e^{-0.004 r} \\
\hline
\end{array}$$
Which country has a decreasing population? By what percentage is the population of that country decreasing each year?

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Key Concepts

Exponential Growth and Decay

Exponential models describe how quantities change over time at a rate proportional to their current value. When an exponential function has a positive exponent, it indicates growth, while a negative exponent indicates decay. These models are widely used to represent processes like population changes, radioactive decay, and interest accumulation.

Continuous Growth Rate Interpretation

In an exponential model of the form A = A?e^(kt), the constant k is the continuous growth rate. A positive k means that the quantity grows continuously, whereas a negative k implies that the quantity decreases continuously. The magnitude of k represents the speed of this change over time.

Conversion of the Continuous Rate to Percentage Change

The continuous rate k can be expressed as a percentage change per unit time. Specifically, for small values of k, the percentage change is approximately |k| × 100%. In the case of decay (when k is negative), this approximates the percentage by which the quantity decreases each period, aligning with the interpretation of the model parameter.

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