To determine the function that best models the given rental scenario, let's break down the two parts of the cost structure:
1. For rentals less than five hours, the cost is \[tex]$3.45 per hour.
2. For rentals of five or more hours, the cost is \$[/tex]3 per hour, plus a \[tex]$1 processing fee.
Based on these conditions, we can express the total cost \(C(x)\) as a piecewise function of the rental hours \(x\):
- For \(x < 5\), the cost is straightforwardly \(3.45x\).
- For \(x \geq 5\), the cost is given by \(3.00x + 1\) (since it's \$[/tex]3 per hour plus the \[tex]$1 processing fee).
Now, let's examine the given options to see which one correctly represents this scenario:
1. \( C(x) = \left\{ \begin{array}{cc} 3.45x, & x < 5 \\ 3.00x + 1, & x \geq 5 \end{array} \right. \)
2. \( C(x) = \left\{ \begin{array}{rr} 3.45x, & x < 5 \\ 3.00 + x, & x > 5 \end{array} \right. \)
3. \( C(x) = \left\{ \begin{array}{cc} 3.45x, & x \leq 4 \\ 3.00x - 1, & x > 5 \end{array} \right. \)
4. \( C(x) = \left\{ \begin{array}{cc} 3.45x, & x < 4 \\ 3.00x - 1, & x \geq 5 \end{array} \right. \)
Let's match these options with the given conditions:
- \( C(x) = \left\{ \begin{array}{cc}
3.45x, & x < 5 \\
3.00x + 1, & x \geq 5
\end{array} \right. \)
This option correctly represents the costs for both conditions. For \(x < 5\), it charges \(3.45x\). For \(x \geq 5\), it charges \(3.00x\) plus a \$[/tex]1 fee.
- [tex]\( C(x) = \left\{ \begin{array}{rr}
3.45x, & x < 5 \\
3.00 + x, & x > 5
\end{array} \right. \)[/tex]
This option is incorrect because it simplifies the second part incorrectly. It implies adding 3.00 and [tex]\(x\)[/tex], which doesn't align with the cost structure.
- [tex]\( C(x) = \left\{ \begin{array}{cc}
3.45x, & x \leq 4 \\
3.00x - 1, & x > 5
\end{array} \right. \)[/tex]
This option is incorrect because the conditions split at [tex]\(x \leq 4\)[/tex] and does not appropriately cover next range.
- [tex]\( C(x) = \left\{ \begin{array}{cc}
3.45x, & x < 4 \\
3.00x - 1, & x \geq 5
\end{array} \right. \)[/tex]
This option is incorrect for similar reasons as the third, with an additional incorrect pricing formula.
Therefore, the correct function that models the scenario is:
[tex]\[ C(x) = \left\{ \begin{array}{ll}
3.45x, & x < 5 \\
3.00x + 1, & x \geq 5
\end{array} \right. \][/tex]
