The population of a colony of mosquitoes obeys the law of uninhibited growth. Use this information to answer parts (a) through (c).
(a) If $\mathrm{N}$ is the population of the colony and $\mathrm{t}$ is the time in days, express $\mathrm{N}$ as a function of $\mathrm{t}$. Consider $\mathrm{N}_{0}$ is the original amount at $\mathrm{t}=0$ and $\mathrm{k} \neq 0$ is a constant that represents the growth rate.
\[
N(t)=N_{0} e^{k t}
\]
(Type an expression using $t$ as the variable and in terms of $e$.)
(b) The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1400 after 1 day, what is the size of the colony after 4 days?

Approximately $\square$ mosquitoes
(Do not round until the final answer. Then round to the nearest whole number as needed.)

Step 1 ：The population of a colony of mosquitoes obeys the law of uninhibited growth, which means that the population grows exponentially over time. The general form of an exponential growth function is given by: \(N(t) = N0 * e^{kt}\), where \(N(t)\) is the population at time \(t\), \(N0\) is the initial population (at time \(t = 0\)), \(k\) is the growth rate, and \(t\) is the time.

Step 2 ：We are given that the initial population (\(N0\)) is 1000 mosquitoes and that the population after 1 day (\(N(1)\)) is 1400 mosquitoes. We can use these values to find the growth rate (\(k\)).

Step 3 ：First, we substitute \(N0 = 1000\) and \(N(1) = 1400\) into the exponential growth function: \(1400 = 1000 * e^{k*1}\).

Step 4 ：Next, we solve for \(k\): \(e^k = 1400 / 1000 = 1.4\). Taking the natural logarithm of both sides gives: \(k = \ln(1.4) \approx 0.3365\).

Step 5 ：Now that we have the growth rate, we can find the population after 4 days (\(N(4)\)) by substituting \(N0 = 1000\), \(k = 0.3365\), and \(t = 4\) into the exponential growth function: \(N(4) = 1000 * e^{0.3365*4}\).

Step 6 ：Calculating this gives: \(N(4) \approx 1000 * e^{1.346} \approx 3842.8\). Rounding to the nearest whole number gives approximately 3843 mosquitoes.

Step 7 ：So, the population of the mosquitoes after 4 days will be \(\boxed{3843}\) mosquitoes.