The population of a city is modeled by the expression [tex]40,000(1.1)^t[/tex], where [tex]t[/tex] represents the number of years since the population was 40,000. Which expression is equivalent and uses a growth rate representing the approximate monthly increase in the city's population?

A. [tex]40,000(1.008)^t[/tex]
B. [tex]40,000(1.008)^{12t}[/tex]
C. [tex]40,000(1.1)^{12t}[/tex]
D. [tex]40,000(1.1)^t[/tex]



Answer :

To determine which expression is equivalent to the given population growth model [tex]\( 40,000(1.1)^t \)[/tex] but represents the growth on a monthly basis, we need to convert the annual growth rate to a monthly growth rate.

Here's a detailed step-by-step solution:

1. Identify the annual growth rate:
The given expression [tex]\( 40,000(1.1)^t \)[/tex] shows that the population increases by a factor of [tex]\( 1.1 \)[/tex] each year. So, the annual growth rate is [tex]\( 1.1 \)[/tex].

2. Convert the annual growth rate to a monthly growth rate:
To find the equivalent monthly growth rate, we need to determine the rate [tex]\( r \)[/tex] such that when compounded 12 times (for 12 months in a year), it results in the same annual growth multiplier [tex]\( 1.1 \)[/tex]. This can be done using the following relationship:
[tex]\[ (1 + \text{monthly\_rate})^{12} = 1.1 \][/tex]
Solving for the monthly rate:
[tex]\[ \text{monthly\_rate} = (1.1)^{1/12} \approx 1.008 \][/tex]

3. Express the population model using the monthly growth rate:
We now rewrite the original population model using the monthly growth rate. If [tex]\( t \)[/tex] is the number of years, then [tex]\( 12t \)[/tex] is the number of months. Therefore, the equivalent expression in terms of months is:
[tex]\[ 40,000 \times (1.008)^{12t} \][/tex]

4. Choose the correct option:
Out of the given options, the expression that is equivalent to [tex]\( 40,000(1.1)^t \)[/tex] but uses the monthly growth rate is:
[tex]\[ \boxed{40,000(1.008)^{12t}} \][/tex]

Thus, the correct answer is B. [tex]\( 40,000(1.008)^{12t} \)[/tex].