To model the growth of the bacteria colony, we can use the exponential growth formula:

$A(t) = A_0 e^{rt}$

Where:

• $A(t)$ is the population after time $t$ (in hours),
• $A_0$ is the initial population (2,500 bacteria),
• $r$ is the growth rate per hour (expressed as a decimal),
• $t$ is the time in hours.

Step 1: Identify the parameters

• Initial population $A_0 = 2500$,
• Growth rate $r = 15\% = 0.15$ per hour.

So, the exponential growth model becomes:

$A(t) = 2500 \times e^{0.15t}$

Step 2: Find the population after 12 hours

Now, substitute $t = 12$ into the equation:

$A(12) = 2500 \times e^{0.15 \times 12}$ $A(12) = 2500 \times e^{1.8}$

Using $e^{1.8} \approx 6.0496$:

$A(12) \approx 2500 \times 6.0496$ $A(12) \approx 15,124$

Thus, the number of bacteria after 12 hours is approximately 15,124.

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Would you like further details or have any other questions? Here are five related questions:

1. What happens to the bacteria population after 24 hours?
2. How would the equation change if the growth rate was compounded continuously at 10%?
3. How long will it take for the population to double in size?
4. What is the population after 6 hours instead of 12?
5. How would the model change if the bacteria die off at a rate of 5% per hour?

Tip: Exponential growth processes are often faster than linear ones—small percentage increases can lead to big changes over time!