Question

question 37 of 40
the population of a country was 954,000 in 2009 with an annual growth rate of 0.05%.
(a) find a mathematical model that relates the population of a country as a function of the number of years after 2009.
(b) if the annual rate of increase remains the same, use this model to predict the population of a country in the year 2050. round to the nearest thousand.
part 1 of 2
(a) the model is $p(t) = $
part 2 of 2
(b) the population in the year 2050 will be approximately

Explanation:

Part (a)

Step1: Recall Exponential Growth Formula

The general formula for exponential growth is \( P(t) = P_0(1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the annual growth rate (in decimal), and \( t \) is the time in years.

Step2: Identify Values

Here, \( P_0 = 954000 \), \( r = 0.05\% = 0.0005 \) (converted to decimal by dividing by 100).

Step3: Substitute into Formula

Substitute \( P_0 \) and \( r \) into the formula: \( P(t) = 954000(1 + 0.0005)^t = 954000(1.0005)^t \).

Step1: Calculate Time \( t \)

First, find the number of years from 2009 to 2050. \( t = 2050 - 2009 = 41 \) years.

Step2: Substitute \( t = 41 \) into Model

Use the model \( P(t) = 954000(1.0005)^{41} \). Calculate \( (1.0005)^{41} \approx 1.0207 \) (using a calculator). Then \( P(41) = 954000 \times 1.0207 \approx 954000 \times 1.0207 \approx 973747.8 \).

Step3: Round to Nearest Thousand

Rounding 973747.8 to the nearest thousand gives 974000.

Answer:

\( 954000(1.0005)^t \)

Part (b)