Let's analyze and solve the problem based on the image you've provided. We're tasked with calculating the average rate of change of bacteria in two different time intervals based on the table of values.

The formula for the average rate of change is: $\text{Average rate of change} = \frac{\Delta N(t)}{\Delta t} = \frac{N(t_2) - N(t_1)}{t_2 - t_1}$ Where $N(t_1)$ and $N(t_2)$ are the number of bacteria at times $t_1$ and $t_2$, respectively.

(a) Average rate of change from 3.1 to 6.2 hours:

From the table:

• $t_1 = 3.1$ hours, $N(t_1) = 1589$
• $t_2 = 6.2$ hours, $N(t_2) = 2612$

Using the formula: $\text{Average rate of change} = \frac{2612 - 1589}{6.2 - 3.1} = \frac{1023}{3.1} \approx 329.68 \, \text{bacteria per hour}$

(b) Average rate of change from 12.4 to 15.5 hours:

From the table:

• $t_1 = 12.4$ hours, $N(t_1) = 4100$
• $t_2 = 15.5$ hours, $N(t_2) = 6425$

Using the formula: $\text{Average rate of change} = \frac{6425 - 4100}{15.5 - 12.4} = \frac{2325}{3.1} \approx 750.00 \, \text{bacteria per hour}$

Summary of results:

• (a) The average rate of change from 3.1 to 6.2 hours is approximately 329.68 bacteria per hour.
• (b) The average rate of change from 12.4 to 15.5 hours is 750 bacteria per hour.

Would you like further clarification on any part of the solution?

Here are 5 related questions to consider:

1. What does the average rate of change tell us about the bacterial growth between two times?
2. How can you calculate the average rate of change using different intervals of time?
3. Is the bacterial growth rate constant over time based on this table? Why or why not?
4. How would the average rate of change differ if the time intervals were smaller?
5. What are some potential real-life factors that could affect bacterial growth rates?

Tip: When calculating average rates of change, always carefully check that your chosen time intervals match the problem's requirements!