Answered

A telemarketer earns a base salary of $3,000 per month, plus an annual bonus of 2.5% of total sales above $250,000. Which function represents the annual salary for total sales of $x$ dollars, where $x$ is greater than $250,000?


A$S(x) = 3,000 + 0.025(x - 250,000)$
B$S(x) = 3,000 + 2.5(x - 250,000)$
C$S(x) = 12(3,000) + 0.025(x - 250,000)$
D$S(x) = 12(3,000) + 2.5(x - 250,000)$

Verified Answer :

  • The annual base salary is calculated as \(12 \times \$3,000 = \$36,000\).
  • The bonus is \(2.5\%\) of sales exceeding \(\$250,000\), represented as \(0.025(x - 250,000)\).
  • Combining these components, the total annual salary function is \(S(x) = 12(3,000) + 0.025(x - 250,000)\).
  • The correct option is \(\boxed{S(x) = 12(3,000) + 0.025(x - 250,000)}\).

Explanation

  1. Determining the Annual Salary Function

To find the annual salary \(S(x)\), we must consider the two components of the telemarketer's earnings: the base salary and the annual bonus.

  1. Base Salary: The telemarketer earns a monthly base salary of \(\$3,000\). Since there are 12 months in a year, the total annual base salary is:

\[12 \times \$3,000 = \$36,000\]

  1. Annual Bonus: The telemarketer receives an annual bonus of \(2.5\%\) on sales exceeding \(\$250,000\).
  • Convert the percentage \(2.5\%\) to a decimal: \(2.5 / 100 = 0.025\).
  • The amount of sales exceeding \(\$250,000\) is given by \((x - 250,000)\).
  • Therefore, the annual bonus is calculated as \(0.025 \times (x - 250,000)\).
  1. Total Annual Salary: Summing these two components gives the total annual salary function \(S(x)\):

\[S(x) = 36,000 + 0.025(x - 250,000)\]
Comparing this to the given options, we see that \(36,000\) is equivalent to \(12 \times 3,000\), so the function is \(S(x) = 12(3,000) + 0.025(x - 250,000)\).

Examples

This type of mathematical modeling is crucial in real-world business environments for compensation structures. By representing complex salary schemes as a single function, companies can easily estimate labor costs based on sales projections, and employees can visualize how their performance directly impacts their total annual earnings. Understanding such models helps in making informed financial decisions regarding career choices and performance goal setting.

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