A country's population and the growth rate in 2011 are given below. Use this information and an exponential model to estimate the country's population in the specified year, population $$ =156 $$ million; growth rate $$ =1.589 $$ percent; year, 2020 The size of the population in the 2020 will be about 180.26 million. (Round to the nearest hundredth as needed.)

Question

UpStudy AI · Accepted Answer

To estimate the country's population in 2020 using an exponential growth model, we can use the formula:

$$
P(t) = P_0 \cdot e^{rt}
$$

where:
- $$ P(t) $$ is the population at time $$ t $$,
- $$ P_0 $$ is the initial population,
- $$ r $$ is the growth rate (as a decimal),
- $$ t $$ is the time in years,
- $$ e $$ is the base of the natural logarithm (approximately equal to 2.71828).

### Step 1: Identify the known values
- Initial population $$ P_0 = 156 $$ million
- Growth rate $$ r = 1.589\% = \frac{1.589}{100} = 0.01589 $$
- Time $$ t = 2020 - 2011 = 9 $$ years

### Step 2: Substitute the values into the formula
Now we can substitute the known values into the formula:

$$
P(9) = 156 \cdot e^{0.01589 \cdot 9}
$$

### Step 3: Calculate the exponent
First, we calculate the exponent:

$$
0.01589 \cdot 9 = 0.14301
$$

### Step 4: Calculate $$ e^{0.14301} $$
Now we calculate $$ e^{0.14301} $$:

$$
e^{0.14301} \approx 1.1534
$$

### Step 5: Calculate the population
Now we can calculate the population in 2020:

$$
P(9) = 156 \cdot 1.1534 \approx 180.26
$$

### Final Result
Thus, the estimated population in 2020 is approximately $$ 180.26 $$ million.

This matches the provided answer, confirming that the calculation is correct.
The estimated population in 2020 is approximately 180.26 million.

Upstudy AI Solution

Answer

Solution

Step 1: Identify the known values

Step 2: Substitute the values into the formula

Step 3: Calculate the exponent

Step 4: Calculate

Step 5: Calculate the population

Final Result

Beyond the Answer

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