To determine the number of months needed until the sales are the same for both headband styles, Sharon needs to create a system of equations that model the sales growth for each style.
Let's examine each style:
1. Style A:
- The initial sales for Style A is 40 headbands.
- The sales increase by [tex]\(10\%\)[/tex] each month. This can be represented by multiplying the sales by [tex]\(1.1\)[/tex] each month.
- Therefore, the sales after [tex]\(m\)[/tex] months can be written as [tex]\(s = 40 \times (1.1)^m\)[/tex].
2. Style B:
- The initial sales for Style B is 20 headbands.
- The sales increase by [tex]\(15\%\)[/tex] each month. This can be represented by multiplying the sales by [tex]\(1.15\)[/tex] each month.
- Therefore, the sales after [tex]\(m\)[/tex] months can be written as [tex]\(s = 20 \times (1.15)^m\)[/tex].
Given these equations, the system of equations that represent the sales growth of both styles is:
[tex]\[ s = 40 \times (1.1)^m \][/tex]
[tex]\[ s = 20 \times (1.15)^m \][/tex]
Upon examining the options, option C correctly represents the system of equations:
[tex]\[ s = 40 \times (1.1)^m \][/tex]
[tex]\[ s = 20 \times (1.15)^m \][/tex]
Thus, the correct answer is:
C. [tex]\(s = 40(1.1)^m\)[/tex]
[tex]\[ s = 20(1.15)^m \][/tex]
This system of equations can be used to determine the number of months, [tex]\(m\)[/tex], until the sales are equal for both headband styles.
