select the correct answer.

an action movie was released two months ago and has been playing at a local movie theater. the number of tickets purchased each week is shown in the table.

which equation best models the relationship between the number of weeks since the movie was released and the number of tickets purchased?

$y = 362x + 27$
$y = 345\sqrt{x - 1} + 82$
$y = 82\sqrt{x - 1} + 345$
$y = 27x + 362$

Question

select the correct answer.

an action movie was released two months ago and has been playing at a local movie theater. the number of tickets purchased each week is shown in the table.

which equation best models the relationship between the number of weeks since the movie was released and the number of tickets purchased?

$y = 362x + 27$
$y = 345\sqrt{x - 1} + 82$
$y = 82\sqrt{x - 1} + 345$
$y = 27x + 362$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Square Root Regression", "Mathematical Modeling" ], "current_concepts": [ "Regression Models", "Scatter Plot Analysis", "Square Root Regression", "Mathematical Modeling" ] } ## Evaluate linear models at boundary points $$ \begin{aligned} &\text{For } y = 362x + 27:\ &x = 1 \implies y = 362(1) + 27 = 389 \quad (\text{Actual: } 348)\ &x = 8 \implies y = 362(8) + 27 = 2923 \quad (\text{Actual: } 545)\ \ &\text{For } y = 27x + 362:\ &x = 1 \implies y = 27(1) + 362 = 389 \quad (\text{Actual: } 348)\ &x = 8 \implies y = 27(8) + 362 = 578 \quad (\text{Actual: } 545) \end{aligned} $$ ## Evaluate square root models at boundary points $$ \begin{aligned} &\text{For } y = 345\sqrt{x-1} + 82:\ &x = 1 \implies y = 345(0) + 82 = 82 \quad (\text{Actual: } 348)\ \ &\text{For } y = 82\sqrt{x-1} + 345:\ &x = 1 \implies y = 82(0) + 345 = 345 \quad (\text{Actual: } 348)\ &x = 2 \implies y = 82(1) + 345 = 427 \quad (\text{Actual: } 402)\ &x = 5 \implies y = 82(2) + 345 = 509 \quad (\text{Actual: } 520)\ &x = 8 \implies y = 82\sqrt{7} + 345 \approx 82(2.646) + 345 \approx 562 \quad (\text{Actual: } 545) \end{aligned} $$ ## Compare model errors and select best fit $$ \begin{aligned} &\text{The model } y = 82\sqrt{x-1} + 345 \text{ yields values closest to the actual data points across the entire domain.}\ &\text{It correctly captures the decreasing rate of growth (concave down behavior) of the ticket sales over time.} \end{aligned} $$

(A) $y = 362x + 27$

(B) $y = 345\sqrt{x - 1} + 82$ (C) $y = 82\sqrt{x - 1} + 345$

(D) $y = 27x + 362$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Regression Models" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Evaluate linear models at boundary points

Evaluate square root models at boundary points

Compare model errors and select best fit

Answer: