1. **Problem statement:** We start with $a$ cells, and the population doubles every 30 minutes. We want to find the formula for $f(t)$, the number of cells after $t$ hours. 2. **Understanding the problem:** Since the cells double every 30 minutes, in 1 hour (which is 60 minutes), the cells double twice. 3. **Formula for exponential growth:** The general formula for exponential growth is: $$f(t) = a \times r^{n}$$ where $a$ is the initial amount, $r$ is the growth rate per time interval, and $n$ is the number of intervals. 4. **Applying to this problem:** Here, the growth rate $r = 2$ (doubling), and the number of intervals $n$ in $t$ hours is: $$n = \frac{t \text{ hours} \times 60 \text{ minutes/hour}}{30 \text{ minutes}} = 2t$$ 5. **Final formula:** Substitute $n = 2t$ into the formula: $$f(t) = a \times 2^{2t}$$ This formula gives the number of cells after $t$ hours, starting from $a$ cells and doubling every 30 minutes.