A bacteria culture is started with 350 bacteria. After 4 hours, the population has grown to 612 bacteria. If the population grows exponentially according to the formula Pt=P0(1+r)t

Given the exponential growth formula:

\[
P_t = P_0(1 + r)^t
\]

Where:
- \( P_t \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the time in hours.

We are given:
- Initial population (\( P_0 \)) = 350,
- Population after 4 hours (\( P_t \)) = 612,
- Time (\( t \)) = 4 hours.

We need to find the growth rate \( r \).

### Step 1: Plug in the known values into the formula.

\[
612 = 350(1 + r)^4
\]

### Step 2: Solve for \( r \).

1. Divide both sides by 350:

\[
\frac{612}{350} = (1 + r)^4
\]

\[
1.7486 \approx (1 + r)^4
\]

2. Take the 4th root of both sides:

\[
(1 + r) = \sqrt[4]{1.7486}
\]

3. Calculate the 4th root:

\[
1 + r \approx 1.148
\]

4. Subtract 1 from both sides to solve for \( r \):

\[
r \approx 0.148
\]

Thus, the growth rate \( r \) is approximately 0.148 or 14.8%.

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Would you like me to show a detailed explanation for each step or help with related questions?

Here are 5 follow-up questions you might consider:
1. How would the population change if the growth rate was higher?
2. What would the population be after 10 hours with the same growth rate?
3. How does exponential growth differ from linear growth?
4. What other real-world phenomena follow an exponential growth model?
5. How can we adjust the formula for different initial populations?

**Tip:** Exponential growth means the population grows faster as time increases because the growth is proportional to the current population!

A bacteria culture is started with 350 bacteria. After 4 hours, the population has grown to 612 bacteria. If the population grows exponentially according to the formula Pt=P0(1+r)^t, what is the growth rate?

Discover how to calculate the exponential growth rate of a bacteria culture that starts with 350 bacteria and grows to 612 after 4 hours. Learn the step-by-step process of solving for the growth rate using the exponential growth formula. Suitable for students in Grades 10-12.

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