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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Cubic Function Graphs", "Average Rate of Change" ], "new_concepts": [], "current_concepts": [ "Cubic Function Graphs", "Average Rate of Change", "Function Comparison" ] } ## Calculate average rate of change for clothing sales $c(x)$ $$ \begin{aligned} &c(2) \approx 130 \ &c(6) \approx 70 \ &\text{Rate of change} = \frac{c(6) - c(2)}{6 - 2} \approx \frac{70 - 130}{4} = -15 \end{aligned} $$ ## Calculate average rate of change for shoes and accessories $s(x)$ $$ \begin{aligned} &s(x) = -1.1x^3 + 10.2x^2 - 32.4x + 127.2 \ &s(2) = -1.1(8) + 10.2(4) - 32.4(2) + 127.2 = -8.8 + 40.8 - 64.8 + 127.2 = 94.4 \ &s(6) = -1.1(216) + 10.2(36) - 32.4(6) + 127.2 = -237.6 + 367.2 - 194.4 + 127.2 = 62.4 \ &\text{Rate of change} = \frac{s(6) - s(2)}{6 - 2} = \frac{62.4 - 94.4}{4} = -8 \end{aligned} $$ ## Compare the rates of change $$ \begin{aligned} &|-8| < |-15| \ &\text{The rate of change for } s(x) \text{ is slower than the rate of change for } c(x). \ &\text{Both rates are negative, meaning the revenues are decreasing.} \end{aligned} $$ Between the 2nd and 6th months, the revenue earned from sales of shoes and accessories is decreasing at a slower rate than the revenue earned from sales of clothing. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Average Rate of Change" ] }

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Explanation:

Calculate average rate of change for clothing sales \(c(x)\)

Calculate average rate of change for shoes and accessories \(s(x)\)

Compare the rates of change

Answer: