The problem in the image asks for the time it will take for the population of bacteria to reach 1 million, given that the bacteria double every hour.

Problem Breakdown:

• Initial bacteria count: 1,000.
• After 1 hour: 2,000 bacteria (this indicates the bacteria double every hour).
• We need to find: The approximate time it will take for the bacteria to reach 1 million.

Solution Approach:

This problem describes exponential growth, where the number of bacteria $N(t)$ after $t$ hours is given by the formula:

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

Where:

• $N_0$ is the initial number of bacteria (1,000),
• $2^t$ represents the doubling every hour,
• $N(t)$ is the number of bacteria after $t$ hours.

We want to find $t$ such that $N(t) = 1,000,000$.

Step-by-Step Calculation:

1. Start with the equation:

   $1,000 \times 2^t = 1,000,000$

2. Divide both sides by 1,000:

   $2^t = 1,000$

3. Take the logarithm of both sides to solve for $t$:

   $t \log(2) = \log(1,000)$

4. Knowing that $\log(2) \approx 0.3010$ and $\log(1,000) = 3$:

   $t = \frac{3}{0.3010} \approx 9.97$

So, the bacteria population will reach 1 million in approximately 10 hours.

Conclusion:

It will take about 10 hours for the bacteria population to reach 1 million, based on exponential growth.

Would you like more details or have any questions about the solution? Here are some related questions to further explore:

1. How would the result change if the bacteria tripled every hour instead of doubling?
2. How can the concept of exponential growth be applied to other real-life scenarios?
3. What happens if the initial population is different, say 500 bacteria? How would that affect the time to reach 1 million?
4. What is the general form of the exponential growth equation for other doubling times (e.g., every half-hour)?
5. How can logarithms help simplify more complex growth problems like this one?

Tip: Exponential growth models are not only useful in biology but also in finance and technology fields (e.g., compound interest or computing power projections).