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].
