A culture of bacteria in a particular dish has an initial population of 200 cells grows at a rate of \( N(t)=40 e^{0.18310} \) cells/day, a. Find the population \( N(t) \) at any time \( t \geq 0 \). b. What is the population after 13 days?

To find the population \( N(t) \), we start with the given growth rate equation. The general solution for exponential growth can be expressed as: \[ N(t) = N_0 e^{kt} \] where \( N_0 \) is the initial population and \( k \) is the growth constant. Since the initial population is 200 cells, we can rewrite the equation with this information. For the provided rate \( N(t) = 40 e^{0.18310t} \), we can clarify the growth model. However, since it seems to relate to the rate of change, we should use: \[ N(t) = 200 e^{0.18310 t} \] This expresses the population at any time \( t \geq 0 \). Next, to find the population after 13 days, substitute \( t = 13 \) into the equation: \[ N(13) = 200 e^{0.18310 \cdot 13} \] Calculating this yields: \[ N(13) \approx 200 e^{2.3843} \approx 200 \times 10.8 \approx 2160 \text{ cells} \] So, the population after 13 days is approximately \(\mathbf{2160}\) cells. Now, let’s spice things up with some fun facts! When considering bacterial growth, did you know that some bacteria can double their population in as little as 20 minutes? This rapid growth can lead to a fascinating race against time, where tiny organisms can take over a dish in record time! Also, the principle of exponential growth isn’t just limited to bacteria; it applies in various fields! For instance, in finance, compound interest can lead to astounding growth over time. Picture a single penny doubling every day—after just 30 days, it would be worth over $5 million!