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 [tex]\(c\)[/tex], where [tex]\(x\)[/tex] 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 [tex]\(s\)[/tex], where [tex]\(x\)[/tex] is the number of months since the beginning of the year.

[tex]\[ s(x) = 0.6 x^3 - 10.2 x^2 + 32.4 x + 127.2 \][/tex]

Use this information to complete the statement.

Question

28-07-2024
Mathematics

Answered

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 [tex]\(c\)[/tex], where [tex]\(x\)[/tex] 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 [tex]\(s\)[/tex], where [tex]\(x\)[/tex] is the number of months since the beginning of the year.

[tex]\[ s(x) = 0.6 x^3 - 10.2 x^2 + 32.4 x + 127.2 \][/tex]

Use this information to complete the statement.

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-28T13:35:10-04:00

Certainly! Let's break down the given function model [tex]\( s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2 \)[/tex] step-by-step:

1. Identifying Variables and Coefficients:
- The function [tex]\( s(x) \)[/tex] represents the revenue earned from sales of shoes and accessories each month, where [tex]\( x \)[/tex] is the number of months since the beginning of the year.
- The coefficients and variables of the terms in the polynomial are:
- [tex]\( 0.6x^3 \)[/tex]: This term suggests a cubic component contributing to the revenue based on the months.
- [tex]\( -10.2x^2 \)[/tex]: This indicates a quadratic component that affects the revenue with a negative impact.
- [tex]\( 32.4x \)[/tex]: This linear term shows a positive linear contribution to the revenue.
- [tex]\( 127.2 \)[/tex]: This is a constant term representing the base amount of revenue regardless of the month.

2. Interpreting Each Term:
- [tex]\( 0.6x^3 \)[/tex]: This term likely represents a significant factor that scales with the cube of the number of months. Since the coefficient is positive, it indicates that the revenue increases as [tex]\( x^3 \)[/tex] increases.
- [tex]\( -10.2x^2 \)[/tex]: The negative sign here indicates a negative contribution to the revenue. As the square of months increases, this term reduces the revenue more significantly.
- [tex]\( 32.4x \)[/tex]: This term suggests a steady increase in revenue as months increase.
- [tex]\( 127.2 \)[/tex]: This constant term ensures that there is a minimum baseline revenue every month, regardless of the fluctuations captured by the other terms.

3. Form of the Polynomial Function:
- The function [tex]\( s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2 \)[/tex] itself is already in its expanded form. The polynomial is presented in standard form with terms ordered by the power of [tex]\( x \)[/tex].

4. Complete Statement:
- Based on the function provided, we can complete the statement: "The revenue earned from sales of shoes and accessories each month is modeled by a cubic polynomial function [tex]\( s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2 \)[/tex]. This model suggests a complex relationship between the revenue and the number of months, with contributions from cubic, quadratic, linear, and constant terms."

In summary, the given function accurately characterizes the monthly revenue from shoes and accessories sales using a cubic polynomial, reflecting various dynamic contributions as the year progresses.