$$ 1 \leftarrow $$ A country's population and the growth rate in 2011 are given below. Assume that the growth rate remains the same from year to year. Use this information and an exponential model to estimate the country's population in the specified year. population $$ =146 $$ million; growth rate $$ =-0.328 $$ percent, year, 2024 The size of the population in the 2024 will be about $$ \square $$ million. (Round to the nearest hundredth as needed.)

Question

UpStudy AI · Accepted Answer

We begin with the exponential growth model

$$
P(t)=P_0\left(1+\frac{r}{100}\right)^t,
$$

where  
$$ P_0 = 146 $$ million is the initial population (in 2011),  
$$ r = -0.328 $$ is the annual growth rate in percent, and  
$$ t $$ is the number of years from 2011.

In 2024, the number of years is

$$
t = 2024 - 2011 = 13.
$$

Since the growth rate is $$-0.328\%$$, the annual multiplier is

$$
1 + \frac{-0.328}{100} = 1 - 0.00328 = 0.99672.
$$

Thus, the population in 2024 is estimated by

$$
P(2024)=146\left(0.99672\right)^{13}.
$$

Now we compute $$\left(0.99672\right)^{13}$$. Using logarithms,

$$
\left(0.99672\right)^{13}=\exp\left(13\ln(0.99672)\right).
$$

Approximating $$\ln(0.99672)$$ (note that for small $$x$$, $$\ln(1-x) \approx -x$$),

$$
\ln(0.99672)\approx -0.00328.
$$

Then

$$
13\ln(0.99672)\approx -0.04264,
$$
$$
\exp(-0.04264)\approx 0.95824.
$$

Multiplying by the initial population,

$$
P(2024)\approx146\cdot0.95824\approx139.88.
$$

Thus, the projected population in 2024 is about

$$
\boxed{139.88} \text{ million.}
$$
The country's population in 2024 will be approximately 139.88 million.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Algebra Questions