1. **Problem Statement:** We have a bacteria population starting at 500 and doubling every 3 hours. We want to write an exponential function to model the population after $t$ hours and find the population after 12 hours. 2. **Formula:** The general form for exponential growth is $$P(t) = P_0 \times a^{\frac{t}{T}}$$ where: - $P(t)$ is the population at time $t$, - $P_0$ is the initial population, - $a$ is the growth factor (how much it multiplies by each period), - $T$ is the time period for one growth cycle. 3. **Apply values:** Here, $P_0 = 500$, $a = 2$ (since it doubles), and $T = 3$ hours. 4. **Write the function:** $$P(t) = 500 \times 2^{\frac{t}{3}}$$ 5. **Calculate population after 12 hours:** $$P(12) = 500 \times 2^{\frac{12}{3}} = 500 \times 2^4 = 500 \times 16 = 8000$$ 6. **Answer:** After 12 hours, there will be 8000 bacteria.