https://file.aiquickdraw.com/mathbot/file/1730475723796cymrirw2.jpg

To solve this, let’s analyze each part of the problem.

1. **Determining the Initial Amount of Bacteria:**
   The function provided is:
   \[
   s(n) = 20 \cdot b^n
   \]
   Here, \( s(n) \) represents the amount of bacteria at time \( n \) hours. The term \( 20 \) is the initial quantity (when \( n = 0 \)), because when \( n = 0 \), \( s(0) = 20 \cdot b^0 = 20 \cdot 1 = 20 \). 

   So, the initial number of bacteria is **20**.

2. **Finding the Hourly Percent Growth Rate:**
   We are given that \( b = 1.85 \). To find the hourly growth rate, we interpret \( b = 1.85 \) as representing a 1.85 times increase each hour. The growth rate \( r \) can be calculated using the formula:
   \[
   b = 1 + \frac{r}{100}
   \]
   Solving for \( r \):
   \[
   1.85 = 1 + \frac{r}{100}
   \]
   \[
   \frac{r}{100} = 0.85
   \]
   \[
   r = 85
   \]
   Thus, the hourly percent growth rate is **85%**.

### Final Answers:
- **Initial amount of bacteria:** 20
- **Hourly percent growth rate:** 85%

Would you like a detailed breakdown of any steps or have further questions?

### Related Questions:
1. How would the model change if the initial bacteria count was different?
2. What would the growth rate be if \( b \) was 2.5 instead?
3. How can exponential growth models apply to other biological processes?
4. What happens to the growth rate if \( b \) is less than 1?
5. How would you find the doubling time given the growth rate?

### Tip:
In exponential functions, the base value reveals the growth or decay factor: if greater than 1, it's growth; if between 0 and 1, it's decay.

A professor is examining a new strain of bacteria. The amount of bacteria can be modeled by the function s(n) = 20 * b^n, where n is the number of hours and b is an unknown positive base.

Based on the model, there were initially ____ bacteria.

If b = 1.85, the hourly percent growth rate of the bacteria would be ____%.

Learn how to determine the initial bacteria count and hourly growth rate for a bacterial growth model s(n) = 20 * b^n. Understand the concept of exponential growth and apply it to calculate an 85% growth rate when b = 1.85. Suitable for high school students in Grades 9-12.

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