During a laboratory experiment, there were periods when the number of bacteria in a sample followed an exponential trend. For example, during the 8 days from June 15 to June 23, the number of bacteria in the sample per day could be modeled by an exponential function, increasing from 21, 132 bacteria on June 15 to 29, 125 bacteria on June 23.
a. What is the 8-day growth factor in the number of bacteria per day during this period?
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b. What is the 1-day growth factor in the number of bacteria per day during this period?
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c. Define a function f that models the number of bacteria per day d days since June 15. Be sure to use proper function notation.
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d. What is the 5-day growth factor for the number of bacteria per day during this period?
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e. What is the 5-day percent change for the number of bacteria per day during this period?

Let's go through each part of the problem step by step:

### Part (a): 8-day Growth Factor
To find the growth factor over an 8-day period, we use the formula for exponential growth:
\[
\text{Growth Factor} = \frac{\text{Final Amount}}{\text{Initial Amount}}
\]

Given:
- Initial amount on June 15 (\(N_0\)) = 21,132 bacteria
- Final amount on June 23 (\(N_8\)) = 29,125 bacteria

\[
\text{8-day Growth Factor} = \frac{29,125}{21,132}
\]

Calculating this gives:
\[
\text{8-day Growth Factor} \approx 1.378
\]

### Part (b): 1-day Growth Factor
To find the daily growth factor (\(r\)) in exponential growth, we take the 8th root of the 8-day growth factor since the growth is exponential.

\[
r = \left( \text{8-day Growth Factor} \right)^{\frac{1}{8}} = \left( \frac{29,125}{21,132} \right)^{\frac{1}{8}}
\]

Calculating this gives:
\[
r \approx 1.040
\]

### Part (c): Function \(f\) Modeling the Bacteria Count
The exponential function that models the bacteria count \(f(d)\) can be defined as:
\[
f(d) = N_0 \cdot r^d
\]

Where:
- \(N_0 = 21,132\) (initial amount on June 15)
- \(r \approx 1.040\) (1-day growth factor)
- \(d\) is the number of days since June 15

So the function is:
\[
f(d) = 21,132 \cdot (1.040)^d
\]

### Part (d): 5-day Growth Factor
To find the growth factor over any period, we use the daily growth factor raised to the number of days:
\[
\text{5-day Growth Factor} = r^5 = (1.040)^5
\]

Calculating this gives:
\[
\text{5-day Growth Factor} \approx 1.217
\]

### Part (e): 5-day Percent Change
The percent change over the 5-day period can be found using the growth factor:
\[
\text{Percent Change} = (\text{Growth Factor} - 1) \times 100\%
\]

For the 5-day period:
\[
\text{Percent Change} = (1.217 - 1) \times 100\% = 21.7\%
\]

### Summary of Answers:
a. The 8-day growth factor is approximately **1.378**.  
b. The 1-day growth factor is approximately **1.040**.  
c. The function modeling the number of bacteria is \( f(d) = 21,132 \cdot (1.040)^d \).  
d. The 5-day growth factor is approximately **1.217**.  
e. The 5-day percent change is approximately **21.7%**.

Would you like more details or have any questions about these steps?

### Related Questions:
1. How does changing the growth factor affect the long-term behavior of the exponential function?
2. If the growth factor was slightly higher, say 1.05, how would that impact the bacteria count over the same period?
3. How would you adjust the function \(f(d)\) if the initial count was measured in a different unit, say in thousands?
4. What would be the bacteria count after 10 days if the current growth trend continues?
5. How can you use logarithms to determine the time it takes for the bacteria count to double?

### Tip:
Understanding exponential growth is crucial, especially in fields like biology, finance, and physics. Small changes in the growth factor can lead to significant differences over time.

During a laboratory experiment, there were periods when the number of bacteria in a sample followed an exponential trend. For example, during the 8 days from June 15 to June 23, the number of bacteria in the sample per day could be modeled by an exponential function, increasing from 21,132 bacteria on June 15 to 29,125 bacteria on June 23.

a. What is the 8-day growth factor in the number of bacteria per day during this period?
b. What is the 1-day growth factor in the number of bacteria per day during this period?
c. Define a function f that models the number of bacteria per day d days since June 15. Be sure to use proper function notation.
d. What is the 5-day growth factor for the number of bacteria per day during this period?
e. What is the 5-day percent change for the number of bacteria per day during this period?

Explore the exponential growth of bacteria over an 8-day period, with calculations for the 8-day and 1-day growth factors, percent change, and a function modeling the bacteria count. The population increased from 21,132 to 29,125 bacteria. Suitable for students in Grades 9-12.

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