The population (in millions) for a country is modeled by

P(t)=340+1.9t−0.002t2
,

where t is the number of years since 2000.
Step 2 of 2 :  What was the average rate of change of population from 2000 to 2011? Round your answer to two decimal places and, if necessary, round any intermediate calculations to six decimal places.

The problem asks to find the **average rate of change** of the population from 2000 to 2011. This can be calculated using the formula:

\[
\text{Average rate of change} = \frac{P(t_2) - P(t_1)}{t_2 - t_1}
\]

where:
- \( t_1 = 0 \) (since 2000 corresponds to \( t = 0 \)),
- \( t_2 = 11 \) (since 2011 corresponds to \( t = 11 \)),
- \( P(t) = 340 + 1.9t - 0.002t^2 \).

### Step 1: Calculate \( P(0) \)
\[
P(0) = 340 + 1.9(0) - 0.002(0)^2 = 340.
\]

### Step 2: Calculate \( P(11) \)
\[
P(11) = 340 + 1.9(11) - 0.002(11)^2.
\]
\[
P(11) = 340 + 20.9 - 0.002(121) = 340 + 20.9 - 0.242 = 360.658.
\]

### Step 3: Calculate the average rate of change
\[
\text{Average rate of change} = \frac{P(11) - P(0)}{11 - 0} = \frac{360.658 - 340}{11} = \frac{20.658}{11} \approx 1.878.
\]

Thus, the average rate of change of the population from 2000 to 2011 is approximately **1.88 million people per year**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. What is the instantaneous rate of change of the population at \( t = 11 \)?
2. How would the average rate of change be different if calculated from 2005 to 2015?
3. What year does the population reach its maximum, based on the given model?
4. How does the quadratic term \( -0.002t^2 \) affect the population growth over time?
5. How could the model be adjusted to account for external factors like migration or birth rate changes?

**Tip:** The average rate of change gives a general trend, but instantaneous rate of change, found using derivatives, provides a more precise snapshot at any given time.

The population (in millions) for a country is modeled by P(t)=340+1.9t−0.002t^2, where t is the number of years since 2000. What was the average rate of change of population from 2000 to 2011? Round your answer to two decimal places and, if necessary, round any intermediate calculations to six decimal places.

This problem demonstrates how to calculate the average rate of change of a population modeled by a quadratic function between 2000 and 2011. The solution involves algebraic manipulation of a quadratic equation and calculating the average rate of change. Suitable for students in Grades 9-12.

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