Using the available​ data, the percent of people in the world who are Internet users can be modeled by the function y=−0.0941x2+3.87x+28.7​, with x representing the number of years after 2010 and y representing the percent.
a. Graph the function for the years 2010 through 2040.
b. Find the year in which the percent is maximized. 
c. If the model is​ valid, what will happen to the percent of people in the world who are Internet users after 2031?

The graph above represents the percentage of internet users in the world from 2010 to 2040, as modeled by the given quadratic function.

### Part (b) Finding the year in which the percent is maximized

The function \( y = -0.0941x^2 + 3.87x + 28.7 \) is a quadratic equation, and since the coefficient of \( x^2 \) is negative, the parabola opens downward, meaning there is a maximum point.

To find the year when the percent is maximized, we need to determine the vertex of the parabola. The \( x \)-coordinate of the vertex can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Where \( a = -0.0941 \) and \( b = 3.87 \).

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the year of maximum percentage using the vertex formula:

\[
x = -\frac{b}{2a}
\]
where \( a = -0.0941 \) and \( b = 3.87 \).

Let me know if you have further questions, or you can try running the calculations on your own!

### Part (c) What happens to the percent after 2031?

To see what happens after 2031, observe that since the function is a downward-opening parabola, the percent of internet users increases until it reaches the maximum and then starts decreasing. Since 2031 is close to the vertex (year of maximum), after 2031, the percentage of internet users will begin to decline.

Let me know if you'd like additional help or details!

Here are 5 related questions to expand on the topic:
1. How do we interpret a negative coefficient for \(x^2\) in a quadratic model?
2. How is the vertex formula useful for maximizing or minimizing functions?
3. Can the model be used to predict internet usage in future decades beyond 2040?
4. How does the shape of the parabola relate to real-world trends in internet usage?
5. What factors could cause the internet user percentage to decrease after reaching its maximum?

Tip: For maximum or minimum points, the vertex formula is always a reliable way to find the turning point in a quadratic function.

Using the available​ data, the percent of people in the world who are Internet users can be modeled by the function y=−0.0941x^2+3.87x+28.7​, with x representing the number of years after 2010 and y representing the percent. a. Graph the function for the years 2010 through 2040. b. Find the year in which the percent is maximized. c. If the model is​ valid, what will happen to the percent of people in the world who are Internet users after 2031?

Explore how the percentage of people in the world who use the Internet can be modeled by a quadratic function from 2010 to 2040. Learn how to find the year when the percentage is maximized and what happens to the percentage after 2031. Suitable for Grades 9-12 students studying quadratic functions and parabolas.

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