To find the function that represents the annual salary for total sales of [tex]\( x \)[/tex] dollars, where [tex]\( x \)[/tex] is greater than \[tex]$250,000, we need to consider both the base salary and the additional bonus.
1. Base Salary Calculation:
- The telemarketer earns a base salary of \$[/tex]3,000 per month.
- The annual base salary is calculated by multiplying the monthly salary by 12.
[tex]\[
\text{Annual Base Salary} = 12 \times 3000 = \$36,000
\][/tex]
2. Bonus Calculation:
- The telemarketer earns an annual bonus which is 2.5% of total sales above \[tex]$250,000.
- To calculate the bonus, first determine the amount of sales above \$[/tex]250,000. This is given by [tex]\( x - 250,000 \)[/tex].
- The bonus is 2.5% of this amount. In decimal form, 2.5% is 0.025.
[tex]\[
\text{Bonus} = 0.025 \times (x - 250,000)
\][/tex]
3. Total Annual Salary Calculation:
- The total annual salary [tex]\( S(x) \)[/tex] is the sum of the annual base salary and the bonus.
[tex]\[
S(x) = \text{Annual Base Salary} + \text{Bonus}
\][/tex]
- Substituting the values we calculated:
[tex]\[
S(x) = 36,000 + 0.025 \times (x - 250,000)
\][/tex]
Thus, the function representing the annual salary for total sales of [tex]\( x \)[/tex] dollars is:
[tex]\[
S(x) = 12(3,000) + 0.025(x - 250,000)
\][/tex]
Therefore, the correct function is:
[tex]\[
S(x) = 12(3,000) + 0.025(x - 250,000)
\][/tex]
