Part 1:

A population $P(t)$ of $260$ bacteria increases by $7\%$ every $3$ hours. Let $t$ be time in hours.
The formula is going to look something like: $P(t) = ab^{t}\text{.}$ Our goal is to find values for $a$ and $b\text{.}$
1. What is the initial size of the population? This is the result we should get at $t = 0$ (no time goes by).
$a =$
2. We are told there is an $7 \%$increase that occurs over the span of $3$ hours. We will call this the “overall” percent increase. We want the “hourly” percent increase.
What overall growth factor corresponds to an $7 \%$increase?

Solution: