This is a classic exponential growth problem where the number of bacteria doubles every hour. The formula for exponential growth is:

$N(t) = N_0 \cdot 2^t$

Where:

• $N(t)$ is the number of bacteria after $t$ hours.
• $N_0$ is the initial number of bacteria.
• $t$ is the time in hours.

We are given:

• $N(24) = 1,073,741,824$
• $t = 24$

We need to find $N_0$. Rearranging the formula to solve for $N_0$:

$N_0 = \frac{N(24)}{2^{24}}$

Now, calculate:

$N_0 = \frac{1,073,741,824}{2^{24}} = \frac{1,073,741,824}{16,777,216} = 64$

Thus, the initial number of bacteria was $\boxed{64}$.

Would you like more details on how exponential growth works, or have any other questions?

Here are 5 related questions you might explore:

1. How does doubling time affect the growth rate in exponential functions?
2. If the bacteria doubled every 2 hours instead, what would the population be after 24 hours?
3. What if the initial number of bacteria was 100, how many bacteria would there be after 24 hours?
4. How long would it take for a bacteria population of 500 to reach 1,073,741,824 with the same growth rate?
5. Can you calculate the population after 10 hours, given an initial population of 64?

Tip: Exponential growth is incredibly fast! Small initial numbers can lead to very large populations quickly with doubling.