The total annual enrollment (in millions) in a country's elementary and secondary schools was recorded every three years from 1975 to 1999. The data are shown in the following table:

[tex]
\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Year & 1975 & 1978 & 1981 & 1984 & 1987 & 1990 & 1993 & 1996 & 1999 \\
\hline
Enrollment (in millions) & 60 & 57.4 & 55.9 & 54.2 & 54.1 & 54.9 & 56.7 & 59.1 & 62.7 \\
\hline
\end{tabular}
\]
[/tex]

The following quadratic model predicts the enrollment [tex] y [/tex], in millions, at a time [tex] x [/tex] years after 1975:

[tex]
\[ y = 0.0497x^2 - 1.0925x + 60.1867 \]
[/tex]

Why was a quadratic function chosen to model the enrollment data?

A. A quadratic function was chosen because the enrollment stays constant over time.
B. A quadratic function was chosen because the enrollment is increasing by a constant amount each year after 1975.
C. A quadratic function was chosen because the enrollment is increasing more and more rapidly each year after 1975.
D. A quadratic function was chosen because the enrollment decreases and then increases between 1975 and 1999.

Use the model to predict the enrollment (in million students) in 1984. (Round your answer to one decimal place.)
[tex] \square [/tex] million students

How does the predicted answer from the model compare to the actual enrollment in 1984? (Round to one decimal place if needed.)
The model [tex]\(\square\)[/tex] -Select- the enrollment by [tex]\(\square\)[/tex] million students.

1) The reason for choosing a quadratic function was D. A quadratic function was chosen because the enrollment decreases and then increases between 1975 and 1999.

2) Rounding to one decimal place, the predicted enrollment in 1984 is 54.4 million students.

3) The model slightly overestimates the enrollment by 0.2 million students.

1) A quadratic function was chosen because the enrollment decreases and then increases between 1975 and 1999. This pattern is characteristic of a quadratic function, which can model data that initially decreases and then increases over time (Option D).

2) To predict the enrollment in 1984 using the quadratic model, we need to calculate y when x = 9 (since 1984 is 9 years after 1975):

y = 0.0497(9)² - 1.0925(9) + 60.1867

Let’s compute this as follows:

0.0497 x 81 = 4.0257

1.0925 x 9 = 9.8325

4.0257 - 9.8325 + 60.1867 = 54.3799

3) The actual enrollment in 1984 was 54.2 million students. Comparing the predicted value to the actual value:

Actual value = 54.2 million

Predicted value = 54.4 million

Overestimation = (54.4 - 54.2) 0.2 million students.

franklindozza005 franklindozza005 · Answer 2 · 2023-11-17T19:14:42-05:00

1) D. Adecreasedquadratic function was chosen because the enrollment decreases and then increased between 1975 and 1999.

2) The predicted enrollment in 1984 is 54.4 million students.

3) The model overestimates the enrollment by 0.2 million students.

1) This pattern is characteristic of a quadratic function, which can model data that initially decreases and then increases over time.

2) We need to calculate y when x = 9 (since 1984 is 9 years after 1975):

y = 0.0497(9)² - 1.0925(9) + 60.1867

Compute this as follows:

0.0497 x 81 = 4.0257

1.0925 x 9 = 9.8325

4.0257 - 9.8325 + 60.1867

= 54.3799

3) Comparing the predicted value to the actual value:

Actual value = 54.2 million

Predicted value = 54.4 million

Overestimation = (54.4 - 54.2)

= 0.2 million students.