A colony of bacteria is increasing at the rate of 15% each hour. There are 2,500 bacteria in the colony at the time observations begin. Find an exponential growth model for A, the number of bacteria t hrs after the first observation.

Use the model to determine the number of bacteria in the colony 12 hrs after the initial observation. (Round your answer to the nearest whole number.)

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**.

---

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!

A colony of bacteria is increasing at the rate of 15% each hour. There are 2,500 bacteria in the colony at the time observations begin. Find an exponential growth model for A, the number of bacteria t hrs after the first observation. Use the model to determine the number of bacteria in the colony 12 hrs after the initial observation. (Round your answer to the nearest whole number.)

Learn how to model the exponential growth of a bacteria colony increasing at a rate of 15% per hour, starting with 2,500 bacteria. Find out how many bacteria there will be after 12 hours, using the formula A(t) = A_0 * e^(rt). Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation