Question $6$

E$.$ Coli Infection Escherichia coli $($E$.$ coli$)$ is a bacterium that can reproduce at an exponential rate. The E$.$ coli reproduce by dividing. A small number of E$.$ coli bacteria in the large intestine of a human can trigger a serious infection within a few hours. Consider a particular E$.$ coli infection that starts with $100$ E$.$ coli bacteria. Each bacterium splits into parts every half hour. Assuming none of the bacteria die, the size of the E$.$ coli population after t hours is given by $P(t),{1002}^{2t}$ where $0,t,16.$

a$.$ Find $P(3)$ and $P(6)$

b$.$ Find the time, to the nearest tenth of an hour, it takes for the E$.$ coli population to number one billion.