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### The model:
The function \( f(n) = 0.00006n^2 + 0.012n + 7.884 \) represents the percentage of police officers who are women in cities with populations of \( n \) thousand.

#### First part of the question:
Glen Ellyn, Illinois, has a population of 252.72 thousand. We are asked to estimate the percentage of police officers in Glen Ellyn who are women by using the formula.

We substitute \( n = 252.72 \) into the function \( f(n) \) to estimate the percentage:
\[
f(252.72) = 0.00006(252.72)^2 + 0.012(252.72) + 7.884
\]
I will calculate this now.For Glen Ellyn, Illinois, with a population of 252.72 thousand, the estimated percentage of police officers who are women is approximately **14.7%** (rounded to one decimal place).

#### Second part of the question:
We are asked to find \( n \) when \( f(n) = 13 \), meaning the percentage of women officers is 13%. To do this, we need to solve the equation:
\[
0.00006n^2 + 0.012n + 7.884 = 13
\]
Let's solve for \( n \).The solutions to the equation are \( n \approx -408.7 \) and \( n \approx 208.7 \). Since population cannot be negative, the valid solution is \( n \approx 208.7 \) thousand.

Thus, when the percentage of women police officers is 13%, the city's population is approximately **208.7 thousand** (rounded to one decimal place).

#### What does this mean in the situation?
It means that a city with a population of approximately 208.7 thousand would have about 13% of its police officers being women, based on the given model.

Would you like to go into more detail about any part of this? Here are five questions that could expand the discussion:

1. How would changes in population affect the percentage of women police officers?
2. How can the accuracy of this quadratic model be verified using real data?
3. What might explain the trend in the percentage of women officers as city population increases?
4. How would this model differ if the equation was linear rather than quadratic?
5. What are the limitations of using this model to predict future trends?

**Tip**: When solving a quadratic equation, always consider the context to decide which solution makes sense (positive or negative roots).

The percentages of police officers who are women are listed in the table for various city populations. Given a quadratic model f(n) = 0.00006n^2 + 0.012n + 7.884 for city populations n (in thousands), answer the following questions:

1. Glen Ellyn, Illinois, has a population of 252.72 thousand. Estimate the percentage of police officers in Glen Ellyn who are women.
2. Find n when f(n) = 13 and interpret the meaning.

Learn how to estimate the percentage of women police officers in a city using a quadratic model for city populations. In this example, we calculate for Glen Ellyn, Illinois, and find the population when the percentage of women officers is 13%.

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