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To complete a landscaping project, Adam charges a fee of \$20.00 for his equipment and \$9.50 per hour spent working on the project. To complete this same landscaping project, Caroline charges a fee of \$17.00 for her equipment and \$10.00 per hour spent working on the project. If $x$ represents the number of hours spent working on the landscaping project, what are all the values of $x$ for which Caroline's total charge is greater than Adam's total charge?


A$5 \leq x \leq 6$
B$6 \leq x \leq 7$
C$x<5$
D$x>6$

Verified Answer :

  • Define Adam's charge as \(A(x) = 20 + 9.5x\) and Caroline's charge as \(C(x) = 17 + 10x\).
  • Set up the inequality \(17 + 10x > 20 + 9.5x\).
  • Simplify the expression to find \(0.5x > 3\).
  • Conclude that \(x > 6\), which matches option D.
  • Final answer: \(\boxed{D}\)

Explanation

  1. Defining the Functions

To determine when Caroline's total charge is greater than Adam's total charge, we first model their charges as functions of the number of hours \(x\) spent working on the project. Adam charges a fixed fee of \(\$20.00\) plus \(\$9.50\) per hour, so his total charge is \(A(x) = 20 + 9.5x\). Caroline charges a fixed fee of \(\$17.00\) plus \(\$10.00\) per hour, so her total charge is \(C(x) = 17 + 10x\). We are looking for the values of \(x\) such that \(C(x) > A(x)\).

  1. Solving the Inequality

We set up the inequality \(C(x) > A(x)\) by substituting the expressions we found:
\[17 + 10x > 20 + 9.5x.\]
To solve for \(x\), we first subtract \(9.5x\) from both sides of the inequality:
\[17 + 10x - 9.5x > 20,\]
\[17 + 0.5x > 20.\]
Next, we subtract 17 from both sides to isolate the term with \(x\):
\[0.5x > 20 - 17,\]
\[0.5x > 3.\]
Finally, divide both sides by \(0.5\):
\[x > \frac{3}{0.5} = 6.\]
Thus, Caroline's charge is greater than Adam's charge when the number of hours spent on the project is greater than 6.

  1. Conclusion

Comparing our result \(x > 6\) to the given options:

A\(5 \leq x \leq 6\)
B\(6 \leq x \leq 7\)
C\(x < 5\)
D\(x > 6\)

The inequality \(x > 6\) corresponds to option D.

Examples

This type of problem is common in comparing service providers. For instance, if two contractors offer different pricing models, you can use these linear functions to determine which contractor is more cost-effective depending on the scope of the job. By calculating the intersection point of the costs, you can decide whether a low base fee with a higher hourly rate or a high base fee with a lower hourly rate is more beneficial for the specific duration of your project. Here, Caroline is cheaper for short projects but becomes more expensive once the hours exceed 6.

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