
A scientist puts six bacteria, which multiply at a constant rate, in a Petri dish. She records the number of bacteria each minute thereafter. If she counts 324 bacteria 19 minutes later, at what rate are the bacteria reproducing?




To find the rate at which the bacteria are reproducing after 19 minutes, we can use the same exponential growth formula:

\[
N(t) = N_0 \times r^t
\]

Given:
- \( N_0 = 6 \) (initial number of bacteria),
- \( N(19) = 324 \) (number of bacteria after 19 minutes),
- \( t = 19 \) minutes.

We need to solve for \( r \):

\[
324 = 6 \times r^{19}
\]

First, divide both sides by 6 to isolate \( r^{19} \):

\[
r^{19} = \frac{324}{6} = 54
\]

Now, to solve for \( r \), take the 19th root of both sides:

\[
r = \sqrt[19]{54}
\]

Let's calculate the value of \( r \).The bacteria are reproducing at a rate of approximately \(1.234\) per minute when counted 19 minutes later.

Would you like more details or have any questions about this calculation?

Here are five related questions you might consider:
1. How many bacteria will there be after 25 minutes at this rate?
2. What would the reproduction rate be if there were 400 bacteria after 19 minutes?
3. How long would it take for the bacteria count to reach 500 at this rate?
4. What happens to the rate if the number of bacteria counted after 19 minutes was 500 instead of 324?
5. How would the rate change if the bacteria were counted every 10 minutes instead?

**Tip:** When solving for exponential growth rates, small changes in time or the number of organisms can significantly affect the calculated rate, so precision is crucial.

Learn how to calculate the reproduction rate of bacteria using the exponential growth formula. Solve the problem where a scientist observes 6 bacteria multiplying at a constant rate, reaching 324 bacteria after 19 minutes. Understand the application of exponential growth in biological sciences.

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