The exponential models describe the population of the...

The exponential models describe the population of the indicated country, $A,$ in millions, t years after $2010 .$
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Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?

Key Concepts

Parameter Comparison in Exponential Models

When multiple exponential models are presented, comparing the growth coefficients or factors allows one to identify the model associated with the highest rate of increase. This involves either directly comparing the base factors (if models are in the form P(t) = P? · a?) or converting the continuous growth rates (r in P(t) = P? · e^(rt)) to annual percentage rates. The country with the largest such factor or rate exhibits the greatest annual percentage increase in its population.

Growth Rate and Annual Percentage Increase

The growth rate is a measure of how quickly a population increases over time, expressed as a percentage per year. In an exponential model, the multiplier a in P(t) = P? · a? indicates that the population grows by a factor of a each year; the corresponding percentage increase is (a - 1) · 100%. Alternatively, when using P(t) = P? · e^(rt), the continuous growth rate r can be converted into an annual percentage increase. Comparing these rates across different models helps determine which population is growing faster.

Exponential Growth Models

Exponential growth models describe quantities that increase (or decrease) at a rate proportional to their current value. They are typically written in forms such as P(t) = P? · a? or P(t) = P? · e^(rt), where P? is the initial value, a is the growth factor per time period, and r is the continuous growth rate. These models are useful for describing phenomena like population growth because they capture how a small percentage change accumulates over time.

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