To find the average rate of change of the population from $t = 0$ to $t = 3.5$ hours, we use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{P(t_2) - P(t_1)}{t_2 - t_1}$

Step 1: Identify the Given Values

• $t_1 = 0$ hours, $P(t_1) = 0.01$ grams
• $t_2 = 3.5$ hours, $P(t_2) = 0.17$ grams

Step 2: Apply the Formula

Substitute the given values into the formula: $\text{Average Rate of Change} = \frac{0.17 - 0.01}{3.5 - 0}$ $\text{Average Rate of Change} = \frac{0.16}{3.5}$ $\text{Average Rate of Change} \approx 0.0457 \text{ grams per hour}$

Final Answer:

The average rate of change of the population from 0 to 3.5 hours is approximately 0.0457 grams per hour.

Would you like to explore this concept further or have any additional questions?

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Here are 5 related questions for further exploration:

1. How does the average rate of change relate to the slope of the secant line between two points?
2. What is the significance of the average rate of change in population studies?
3. How would the average rate of change differ if calculated over the entire dataset?
4. Can the average rate of change be negative? If so, what does that imply?
5. How does the instantaneous rate of change differ from the average rate of change?

Tip: Always ensure the units in the numerator and denominator match when calculating rates of change.