your schools art club is organizing a poster design contest where students earn points based on their creativity, technique, and presentation. the event organizers track the total points earned as more students participate in the contest. the model used to predict the total points earned (in points) based on the number of students participating is $\text{total points} = 5n^2 + 2n + 1$, where $n$ is the number of students participating.

please select the appropriate values from the drop down menus for each column. ensure that your choices accurately reflect the predicted revenue and the residuals based on the given model.

recently, you recorded the actual points earned as more students participated:

$\begin{array}{|c|c|c|c|}
\hline
\text{number of students (n)} & \text{measured points} & \text{predicted points} & \text{residual (measured-predicted)} \\
\hline
4 & 90 & \quad & \quad \\
\hline
8 & 350 & \quad & \quad \\
\hline
12 & 750 & 745 & 5 \\
\hline
16 & 1350 & 1313 & 37 \\
\hline
\dots & \dots & & \\
\hline
\end{array}$

Question

your schools art club is organizing a poster design contest where students earn points based on their creativity, technique, and presentation. the event organizers track the total points earned as more students participate in the contest. the model used to predict the total points earned (in points) based on the number of students participating is $\text{total points} = 5n^2 + 2n + 1$, where $n$ is the number of students participating.

please select the appropriate values from the drop down menus for each column. ensure that your choices accurately reflect the predicted revenue and the residuals based on the given model.

recently, you recorded the actual points earned as more students participated:

$\begin{array}{|c|c|c|c|}
\hline
\text{number of students (n)} & \text{measured points} & \text{predicted points} & \text{residual (measured-predicted)} \\
\hline
4 & 90 & \quad & \quad \\
\hline
8 & 350 & \quad & \quad \\
\hline
12 & 750 & 745 & 5 \\
\hline
16 & 1350 & 1313 & 37 \\
\hline
\dots & \dots & & \\
\hline
\end{array}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Quadratic Modeling", "Residual Calculation" ], "current_concepts": [ "Quadratic Modeling", "Residual Calculation" ] } ## Calculate predicted points for n = 4 and n = 8 $$ \begin{aligned} P(n) &= 5n^2 + 2n + 1 \ P(4) &= 5(4)^2 + 2(4) + 1 = 5(16) + 8 + 1 = 80 + 9 = 89 \ P(8) &= 5(8)^2 + 2(8) + 1 = 5(64) + 16 + 1 = 320 + 17 = 337 \end{aligned} $$ ## Calculate residuals for n = 4 and n = 8 $$ \begin{aligned} \text{Residual} &= \text{Measured} - \text{Predicted} \ R(4) &= 90 - 89 = 1 \ R(8) &= 350 - 337 = 13 \end{aligned} $$ ## Verify given values for n = 12 and n = 16 $$ \begin{aligned} P(12) &= 5(12)^2 + 2(12) + 1 = 5(144) + 24 + 1 = 720 + 25 = 745 \quad (\text{Matches}) \ R(12) &= 750 - 745 = 5 \quad (\text{Matches}) \ P(16) &= 5(16)^2 + 2(16) + 1 = 5(256) + 32 + 1 = 1280 + 33 = 1313 \quad (\text{Matches}) \ R(16) &= 1350 - 1313 = 37 \quad (\text{Matches}) \end{aligned} $$ | Number of Students ($n$) | Measured Points | Predicted Points | Residual (measured-predicted) | |---|---|---|---| | 4 | 90 | 89 | 1 | | 8 | 350 | 337 | 13 | | 12 | 750 | 745 | 5 | | 16 | 1350 | 1313 | 37 | { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Modeling" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate predicted points for n = 4 and n = 8

Calculate residuals for n = 4 and n = 8

Verify given values for n = 12 and n = 16

Answer: