Questions: USING TOOLS The table shows the numbers y of students absent from school x days after a flu outbreak. Write a function that models the data. Use the model to approximate the number of students absent 10 days after the outbreak. Time (days), x: 2, 4, 5, 6, 8, 9, 11 Number of students, y: 11, 17, 19, 19, 17, 14, 7

USING TOOLS The table shows the numbers y of students absent from school x days after a flu outbreak. Write a function that models the data. Use the model to approximate the number of students absent 10 days after the outbreak.

Time (days), x: 2, 4, 5, 6, 8, 9, 11
Number of students, y: 11, 17, 19, 19, 17, 14, 7

Transcript text: 4. USING TOOLS The table shows the numbers $y$ of students absent from school $x$ days after a flu outbreak. Write a function that models the data. Use the model to approximate the number of students absent 10 days after the outbreak. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Time (days), $x$ & 2 & 4 & 5 & 6 & 8 & 9 & 11 \\ \hline \begin{tabular}{l} Number of \\ students, $y$ \end{tabular} & 11 & 17 & 19 & 19 & 17 & 14 & 7 \\ \hline \end{tabular}

Solution

Solution Steps

To model the data, we can use polynomial regression, which fits a polynomial equation to the given data points. We will use NumPy's polyfit function to find the coefficients of the polynomial that best fits the data. Then, we will use the polynomial to approximate the number of students absent 10 days after the outbreak.

Step 1: Given Data

We are given the number of students $ y $ absent from school $ x $ days after a flu outbreak: \[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Time (days), } x & 2 & 4 & 5 & 6 & 8 & 9 & 11 \\ \hline \text{Number of students, } y & 11 & 17 & 19 & 19 & 17 & 14 & 7 \\ \hline \end{array} \]

Step 2: Polynomial Regression

We fit a polynomial of degree 2 (quadratic) to the data. The polynomial is of the form: \[ y = ax^2 + bx + c \] The coefficients obtained are: \[ a = -0.4908, \quad b = 5.884, \quad c = 1.357 \] Thus, the polynomial equation is: \[ y = -0.4908x^2 + 5.884x + 1.357 \]

Step 3: Approximate Number of Students Absent

We use the polynomial to approximate the number of students absent 10 days after the outbreak: \[ y(10) = -0.4908(10)^2 + 5.884(10) + 1.357 \] Calculating this, we get: \[ y(10) \approx 11.1237 \]