Problem
Save 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. \[ N(t)=\square \] (Type an expression using $t$ as the variable and in terms of $e_{\text {.) }}$
Solution
Step 1 ：The population of a colony of mosquitoes obeys the law of uninhibited growth. This means that the population grows exponentially over time.
Step 2 ：The general formula for exponential growth is given by: \(N(t) = N_{0} * e^{kt}\) where: \(N(t)\) is the population at time t, \(N_{0}\) is the initial population (i.e., the population at time t=0), \(k\) is the growth rate, and \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
Step 3 ：So, we can express N as a function of t using this formula.
Step 4 ：Final Answer: The population N as a function of time t is given by the formula: \(\boxed{N(t) = N_{0}e^{kt}}\)
