Answer: B. $c = 1600(1.05)^m$

Explanation:1. Identify the initial condition: - In January $2018$, there were $1600$ customers enrolled. This is the initial number of customers when $m=0$ ( $m$ is months *after* January $2018$). So, $c(0) = 1600$.

2. Identify the growth rate: - The total number of customers enrolled each month was $5\%$ greater than the total number enrolled the previous month. - A $5\%$ increase means the new amount is $100\% + 5\% = 105\%$ of the previous month's amount. - As a decimal, $105\% = 1.05$. This is the growth factor.

3. Formulate the exponential growth model: The general form for exponential growth is $c(m) = P(1+r)^m$ or $c(m) = P \cdot (\text{growth factor})^m$, where: - $c(m)$ is the number of customers after $m$ months. - $P$ is the initial number of customers. - $r$ is the growth rate per period (as a decimal). - $m$ is the number of time periods.

In this case, $P = 1600$ and the growth factor is $1.05$. So, the equation is $c = 1600(1.05)^m$.

4. The condition $m \le 24$ indicates the model is valid for the next $24$ months.

5. Compare with the given options: - [A] $c = 1600(0.05)^m$ (This would represent a $95\%$ decrease each month, as $0.05$ is the factor, not $1+0.05$). - [B] $c = 1600(1.05)^m$ (This matches our derived equation for a $5\%$ increase). - [C] $c = 1600(1.5)^m$ (This would represent a $50\%$ increase each month). - [D] $c = 1600(5)^m$ (This would represent a $400\%$ increase each month, or multiplying by $5$).

6. Conclusion: The equation that gives the total number of customers, $c$, enrolled $m$ months after January $2018$ is $c = 1600(1.05)^m$.