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Sharon makes headbands and sells them online. She is analyzing the sales of two of her most popular styles. For style A, she sold 40 in the first month and had a [tex]\(10\%\)[/tex] increase in sales each month after that. For style B, she sold 20 in the first month and had a [tex]\(15\%\)[/tex] increase in sales each month after that.

Which system of equations can she use to determine the number of months, [tex]\(m\)[/tex], until the sales, [tex]\(s\)[/tex], are the same for both headband styles?

A. [tex]\[ s = 40(0.10)^m \][/tex]
[tex]\[ s = 20(0.15)^m \][/tex]

B. [tex]\[ s = 1.1(40)^m \][/tex]
[tex]\[ s = 1.15(20)^m \][/tex]

C. [tex]\[ s = 40(1.1)^m \][/tex]
[tex]\[ s = 20(1.15)^m \][/tex]

D. [tex]\[ s = 10(40)^m \][/tex]
[tex]\[ s = 15(20)^m \][/tex]



Answer :

GinnyAnswer
Let's analyze the sales patterns for both styles of headbands to determine the correct system of equations.

For style A:
- Sharon sold 40 headbands in the first month.
- The sales increase by 10% each month.

To model this mathematically:
- The initial sales are 40.
- With a 10% increase each subsequent month, the sales function can be written as
[tex]\[ s = 40 \times (1 + 0.10)^m = 40 \times (1.1)^m \][/tex]

For style B:
- Sharon sold 20 headbands in the first month.
- The sales increase by 15% each month.

To model this mathematically:
- The initial sales are 20.
- With a 15% increase each subsequent month, the sales function can be written as
[tex]\[ s = 20 \times (1 + 0.15)^m = 20 \times (1.15)^m \][/tex]

Therefore, the system of equations she can use to determine the number of months, [tex]\(m\)[/tex], until the sales, [tex]\(s\)[/tex], are the same for both headband styles is:

[tex]\[ s = 40 \times (1.1)^m \][/tex]
[tex]\[ s = 20 \times (1.15)^m \][/tex]

Thus, the correct answer is:

C. [tex]\( s = 40 \times (1.1)^m \)[/tex]
[tex]\( s = 20 \times (1.15)^m \)[/tex]

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