Suppose a population of bacteria is being grown in a homogenous nutrient medium. It was determined that the population doubled every 30 minutes. If the initial population of the bacteria is given by $P_0$, which of the following functions correctly represents the population $P$ of the bacteria after $t$ hours?

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The only difference between the answer choices is the value of the exponent. Therefore, we just need to figure out what coefficient goes with $t$. Calling this coefficient $a$, we get this general relationship:

$$P(t)=P_0(2)^{at}$$

Because the bacteria population doubles every 30 minutes, we expect the population to double once every half-hour. In other words, the population should double twice every hour. The population at the first hour should be quadruple that of the initial population.

$$P(1)=4P(0) \tag*{\tiny comparing population at \(t=1\) vs \(t=0\)}$$ $$P_0(2)^{a(1)}=4P_0(2)^{a(0)}$$ $$P_0(2)^a=4P_0(2)^{0}$$ $$2^a=4(1)$$ $$a=2$$ $$\boxed{P=P_0(2)^{2t}}$$

Using a similar approach to the first, we can infer that after an hour, the population should double twice. If we were to plug in $t=1$, we should obtain $4P_0$. A quick inspection of the answer choices reveals:

$$\boxed{P=P_0(2)^{2t}}$$ $$P=P_0(2)^{2(1)}$$ $$P=P_0(2)^{2}$$ $$P=P_0(4)$$

None of the other choices would result in the same relationship when $t=1$ is substituted in.