A certain bacteria is known to double every 30 minutes. Suppose there are initially 9 bacteria in a petri dish. Make a table, graph, and an equation.

Time in hours	Number of bacteria
0	
9
1	
36
2	
144
3	
576
4	
2304
5	
9216


9\left(2\right)^{2t}
  

1
2
3
-1
3
6
9
12
15
18
21
24
27
30
33
36
-3
Clear All Draw: Exponential


How many minutes (to the nearest minute) will it take for the bacteria to reach the number 10,000? 
  

How many minutes (to the nearest minute) will it take for the bacteria to reach the number 1,000,000? 
  

How many bacteria will there be after 24 hours? (to the nearest bacteria) 
  

Write a logarithmic equation that would allow you to find the time 
 when there 760 bacteria. 
  

What is the time when there are 760 bacteria? (to the nearest minute) 

To tackle the problem, let's break it down step by step, including constructing the table, graph, equation, and answering the questions provided.

### 1. **Table of Bacteria Growth**

The table you provided shows the bacteria population doubling every 30 minutes (or 0.5 hours). The pattern is exponential, where the number of bacteria at time \( t \) in hours is given by:

\[
N(t) = 9 \times 2^{2t}
\]

This equation works because \( 2t \) accounts for the number of 30-minute intervals in \( t \) hours.

Here's the completed table:

\[
\begin{array}{|c|c|}
\hline
\textbf{Time in hours} & \textbf{Number of bacteria} \\
\hline
0 & 9 \\
1 & 36 \\
2 & 144 \\
3 & 576 \\
4 & 2304 \\
5 & 9216 \\
\hline
\end{array}
\]

### 2. **Graph of the Function**

The graph of the function \( N(t) = 9 \times 2^{2t} \) is an exponential curve. As \( t \) increases, the number of bacteria rises sharply.

If you plot this, you will see the rapid growth as the number of bacteria doubles every half hour.

### 3. **Exponential Equation**

The equation that models the number of bacteria at any time \( t \) in hours is:

\[
N(t) = 9 \times 2^{2t}
\]

### 4. **Answers to Specific Questions**

1. **Time for Bacteria to Reach 10,000:**

   We solve \( 9 \times 2^{2t} = 10000 \).

   \[
   2^{2t} = \frac{10000}{9} \approx 1111.11
   \]
   \[
   2t \log 2 = \log 1111.11
   \]
   \[
   t = \frac{\log 1111.11}{2 \log 2} \approx 5.46 \text{ hours}
   \]

   Converting to minutes:

   \[
   5.46 \times 60 \approx 327.6 \text{ minutes} \approx 328 \text{ minutes}
   \]

2. **Time for Bacteria to Reach 1,000,000:**

   We solve \( 9 \times 2^{2t} = 1000000 \).

   \[
   2^{2t} = \frac{1000000}{9} \approx 111111.11
   \]
   \[
   t = \frac{\log 111111.11}{2 \log 2} \approx 8.94 \text{ hours}
   \]

   Converting to minutes:

   \[
   8.94 \times 60 \approx 536.4 \text{ minutes} \approx 536 \text{ minutes}
   \]

3. **Number of Bacteria After 24 Hours:**

   Substitute \( t = 24 \) into the equation:

   \[
   N(24) = 9 \times 2^{48}
   \]

   \[
   N(24) \approx 2.9 \times 10^{15} \text{ bacteria (to the nearest bacteria)}
   \]

4. **Logarithmic Equation for 760 Bacteria:**

   Starting from \( 9 \times 2^{2t} = 760 \):

   \[
   2^{2t} = \frac{760}{9} \approx 84.44
   \]
   Taking the logarithm:

   \[
   t = \frac{\log 84.44}{2 \log 2}
   \]

5. **Time for 760 Bacteria (to the nearest minute):**

   Solve the equation above:

   \[
   t \approx 3.4 \text{ hours} \times 60 \approx 204 \text{ minutes}
   \]

### Summary of Results:

1. **328 minutes** for 10,000 bacteria.
2. **536 minutes** for 1,000,000 bacteria.
3. **2.9 × 10^15** bacteria after 24 hours.
4. **Logarithmic equation**: \( t = \frac{\log \left(\frac{760}{9}\right)}{2 \log 2} \).
5. **204 minutes** for 760 bacteria.

---

Would you like more details on any of these steps? Here are five related questions to think about:

1. What would be the equation if the bacteria tripled every 30 minutes instead of doubling?
2. How would the graph change if the initial number of bacteria was 20 instead of 9?
3. What happens to the doubling time if the environment changes and slows the growth?
4. Can you determine the rate of growth using a logarithmic scale on the graph?
5. What would be the half-life of the bacteria population if the growth was reversed?

**Tip:** When solving exponential growth problems, converting time to match the growth intervals simplifies the calculations.

A certain bacteria is known to double every 30 minutes. Suppose there are initially 9 bacteria in a petri dish. Make a table, graph, and an equation. How many minutes (to the nearest minute) will it take for the bacteria to reach the number 10,000? How many minutes (to the nearest minute) will it take for the bacteria to reach the number 1,000,000? How many bacteria will there be after 24 hours? (to the nearest bacteria) Write a logarithmic equation that would allow you to find the time when there are 760 bacteria. What is the time when there are 760 bacteria? (to the nearest minute)

Explore the exponential growth of bacteria that doubles every 30 minutes. Learn how to calculate the time it takes for bacteria to reach specific populations, such as 10,000 and 1,000,000. Understand the logarithmic equations needed to find the time for 760 bacteria and the total count after 24 hours. This problem is suitable for students in Grades 9-12.

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