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 countries have a decreasing population? By what percentage is the population of these countries decreasing 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 countries have a decreasing population? By what percentage is the population of these countries decreasing each year?

Key Concepts

Exponential Growth and Decay

Exponential models are used to describe situations where a quantity changes at a rate proportional to its current value. Whether a quantity grows or decays depends on the sign of the exponent: if the constant is positive, the model represents growth; if it is negative, the model represents decay.

Continuous Exponential Models

Continuous exponential models have the form A = A?e^(kt), where A? is the initial amount, k is the continuous growth or decay rate, and t is time. This format is commonly used to model populations, radioactive decay, and other processes that change continuously over time.

Interpreting the Exponent

In the model A = A?e^(kt), the value of k directly determines the behavior of the model. A positive k indicates an increasing quantity (growth), while a negative k indicates a decreasing quantity (decay). Identifying the sign of k is thus crucial in understanding whether the population is increasing or decreasing.

Calculating Annual Percentage Change

For a continuous model, the factor e^(k) represents the multiplicative change over one time unit. To find the annual percentage change, compute (e^(k) - 1) × 100%. In the case of a decay (where k is negative), this result will be a negative percentage indicating the yearly decrease. This conversion allows one to directly interpret the continuous rate as an annual percentage change.

Transcript

00:01 So we're given four models, and we have, one country has a population model of this.

00:13 Now, that would mean that there is growth because that is, the value is positive.

00:18 So where it's growing at 0 .8 % per year, but compounded continuously.

00:24 This next one, 31 .5e to the 0 .019t.

00:31 Again, that is positive.

00:33 So the population is growing 1 .9 % but compounded continuously.

00:39 There's actually more than 1 .9 % if you just look at january 1st to january 1st the following year because of that compounding.

00:48 The next one, you have that beginning part and then we have negative .006 to the t.

00:56 This means a decline.

00:58 The population is decreasing.

01:00 You're losing.

01:01 0 .6 % per year, but it's compounded continuously.

01:06 So it's actually not just a total over the year from january 1st to january 1st the following year of a 0 .6 % decrease.

01:15 And we'll look at that in just a second.

01:17 And then the next equation, 141 .9e to the negative 0 .005t.

01:24 Again, since that's negative, that does mean that the population is decreasing.

01:29 Now, if we take the e to the negative, 0 .006.

01:33 Let's get what that is as a decimal...

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