The exponential models describe the population of the...

The exponential models describe the population of the indicated country, $A,$ in millions, t years after $2010 .$ Use these models to solve Exercises $1-6 .$ $$ \begin{array}{l}{A=1173.1 e^{0.008 t}} \\ {A=31.5 e^{0.019 t}} \\ {A=127.3 e^{-0.006 t}} \\ {A=141.9 e^{-0.005 t}}\end{array} $$ Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?

The exponential models describe the population of the indicated country, $A,$ in millions, t years after $2010 .$ Use these models to solve Exercises $1-6 .$
$$
\begin{array}{l}{A=1173.1 e^{0.008 t}} \\ {A=31.5 e^{0.019 t}} \\ {A=127.3 e^{-0.006 t}} \\ {A=141.9 e^{-0.005 t}}\end{array}
$$
Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?

Key Concepts

Exponential Growth Models

Exponential growth models are mathematical representations of processes that increase (or decrease) at a rate proportional to their current value. They are typically expressed in the form A = A?e^(rt), where A is the amount after time t, A? is the initial amount, and r is the continuous growth rate. This model exemplifies how compounding effects work, yielding rapid increases over time when the growth rate is positive.

Growth Rate

The growth rate in an exponential model is represented by the exponent term, r, which signifies how quickly the quantity increases per unit time. In contexts such as population growth, the value of r directly translates into the speed at which the population grows. A higher positive r indicates a faster rate of increase over time, while a negative r would denote a decreasing trend.

Percentage Increase per Time Unit

In exponential growth, the continuous growth rate r can be interpreted as a percentage increase per time unit after converting it from its decimal form to a percentage. This is done by multiplying r by 100. For instance, a growth rate of 0.019 corresponds to a 1.9% increase per year, meaning that each year the population is increasing by 1.9% of its current size.

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