Question

a town has a population of 10000 and grows at 5% every year. what will be the population after 5 years, to the nearest whole number?

Explanation:

Step1: Recall the compound growth formula

The formula for compound growth is \( P = P_0(1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate (in decimal), and \( t \) is the time in years.
Here, \( P_0 = 10000 \), \( r = 0.05 \) (since 5% = 0.05), and \( t = 5 \).

Step2: Substitute the values into the formula

Substitute \( P_0 = 10000 \), \( r = 0.05 \), and \( t = 5 \) into the formula:
\( P = 10000(1 + 0.05)^5 \)

Step3: Calculate \( (1 + 0.05)^5 \)

First, calculate \( 1 + 0.05 = 1.05 \). Then, \( 1.05^5 \approx 1.27628 \) (using a calculator or by successive multiplication: \( 1.05\times1.05 = 1.1025 \), \( 1.1025\times1.05 = 1.157625 \), \( 1.157625\times1.05 = 1.21550625 \), \( 1.21550625\times1.05 \approx 1.2762815625 \))

Step4: Calculate the population \( P \)

Multiply the initial population by the growth factor:
\( P = 10000\times1.2762815625 \approx 12762.815625 \)

Step5: Round to the nearest whole number

Rounding \( 12762.815625 \) to the nearest whole number gives 12763.

Answer:

12763