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.