select the correct answer from each drop-down menu.

the revenue that an apparel company earns each month for its different product lines fluctuates throughout the year. over the course of a year, the revenue earned from clothing sales each month is modeled by function (c), where (x) is the number of months since the beginning of the year.

the revenue earned from sales of shoes and accessories each month is modeled by function (s), where (x) is the number of months since the beginning of the year.

(s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2)

use this information to complete the statement.

between the 3rd and 6th months, the revenue earned from sales of shoes and accessories is choose an answer at choose an answer the revenue earned from sales of clothing.

Question

select the correct answer from each drop-down menu.

the revenue that an apparel company earns each month for its different product lines fluctuates throughout the year. over the course of a year, the revenue earned from clothing sales each month is modeled by function (c), where (x) is the number of months since the beginning of the year.

the revenue earned from sales of shoes and accessories each month is modeled by function (s), where (x) is the number of months since the beginning of the year.

(s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2)

use this information to complete the statement.

between the 3rd and 6th months, the revenue earned from sales of shoes and accessories is choose an answer at choose an answer the revenue earned from sales of clothing.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Cubic Function Graphs" ], "new_concepts": [ "Function Comparison" ], "current_concepts": [ "Cubic Function Graphs", "Function Comparison" ] } ## Evaluate the function $s(x)$ at the boundaries $x = 3$ and $x = 6$ $$ \begin{aligned} s(3) &= 0.6(3)^3 - 10.2(3)^2 + 32.4(3) + 127.2 \ &= 0.6(27) - 10.2(9) + 97.2 + 127.2 \ &= 16.2 - 91.8 + 97.2 + 127.2 = 148.8 \ s(6) &= 0.6(6)^3 - 10.2(6)^2 + 32.4(6) + 127.2 \ &= 0.6(216) - 10.2(36) + 194.4 + 127.2 \ &= 129.6 - 367.2 + 194.4 + 127.2 = 84.0 \end{aligned} $$ ## Compare $s(x)$ and $c(x)$ on the interval $[3, 6]$ $$ \begin{aligned} &\text{From the graph of } c(x): \ &c(3) \approx 135, \quad c(6) \approx 90 \ &\text{Comparing values at boundaries:} \ &s(3) = 148.8 > c(3) \approx 135 \ &s(6) = 84.0 < c(6) \approx 90 \ &\text{Since } s(3) > c(3) \text{ and } s(6) < c(6), \text{ the curves must intersect on } (3, 6). \end{aligned} $$ ## Determine the correct drop-down statements $$ \begin{aligned} &\text{First drop-down options describe the relationship:} \ &\text{"sometimes greater than and sometimes less than"} \ &\text{Second drop-down options describe the rate of change:} \ &\text{Average rate of change of } s(x) \text{ on } [3, 6]: \ &\frac{s(6) - s(3)}{6 - 3} = \frac{84.0 - 148.8}{3} = -21.6 \ &\text{Average rate of change of } c(x) \text{ on } [3, 6]: \ &\frac{c(6) - c(3)}{6 - 3} \approx \frac{90 - 135}{3} = -15 \ &\text{Comparing rates of change: } -21.6 < -15 \text{ (decreasing at a faster rate).} \end{aligned} $$ Between the 3rd and 6th months, the revenue earned from sales of shoes and accessories is sometimes greater than and sometimes less than the revenue earned from sales of clothing, because the revenue earned from sales of shoes and accessories is decreasing at a faster rate than the revenue earned from sales of clothing. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Function Comparison" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the function \(s(x)\) at the boundaries \(x = 3\) and \(x = 6\)

Compare \(s(x)\) and \(c(x)\) on the interval \([3, 6]\)

Determine the correct drop-down statements

Answer: