After a heavy rainstorm, Barbara's dog left muddy paw prints all over her back patio. She decided to rent a pressure washer to clean up the mess. The piecewise function below shows the rental charges for the washer, dependent upon how many hours she rents it. Due to high demand, you cannot rent the washer for more than 24 hours at a time. Barbara's brother asked to use the washer and agreed to pay Barbara for the time he used it.

[tex]\[ y = \left\{
\begin{array}{ll}
26, & 0 \ \textless \ h \leq 4 \\
26 + 5(h - 4), & 4 \ \textless \ h \leq 8 \\
60, & 8 \ \textless \ h \leq 24
\end{array}
\right. \][/tex]

If Barbara used the washer for 3 hours and her brother returned it to the store 7 hours after she rented it, how much does her brother owe her?

A. \[tex]$15.00
B. \$[/tex]41.00
C. \[tex]$60.00
D. \$[/tex]61.00



Answer :

Sure, let's go through the solution step by step to determine how much Barbara's brother owes her.

### Step 1: Understand the piecewise function for the rental charges
The rental charges [tex]\( y \)[/tex] based on the number of hours [tex]\( h \)[/tex] are given by:
[tex]\[ y = \begin{cases} 26, & 0 < h \leq 4 \\ 26 + 5(h - 4), & 4 < h \leq 8 \\ 60, & 8 < h \leq 24 \end{cases} \][/tex]

### Step 2: Calculate the rental charges for Barbara
Barbara used the washer for 3 hours, meaning [tex]\( h_1 = 3 \)[/tex].
According to the function, since [tex]\( 0 < 3 \leq 4 \)[/tex], the charge for Barbara for 3 hours is:
[tex]\[ y_1 = 26 \][/tex]

### Step 3: Calculate the rental charges for Barbara's brother
Barbara's brother used the washer for 7 hours after Barbara, so in total, he used it for 7 hours. Thus, [tex]\( h_2 = 7 \)[/tex].
According to the function, since [tex]\( 4 < 7 \leq 8 \)[/tex], the charge for 7 hours is:
[tex]\[ y_2 = 26 + 5(7 - 4) \][/tex]
[tex]\[ y_2 = 26 + 5(3) \][/tex]
[tex]\[ y_2 = 26 + 15 \][/tex]
[tex]\[ y_2 = 41 \][/tex]

### Step 4: Calculate the total charges
The total charges for both Barbara and her brother combined are:
[tex]\[ y_{\text{total}} = y_1 + y_2 \][/tex]
[tex]\[ y_{\text{total}} = 26 + 41 \][/tex]
[tex]\[ y_{\text{total}} = 67 \][/tex]

### Step 5: Determine how much Barbara's brother owes her
Since the brother's charge is [tex]\( 41 \)[/tex] and Barbara already paid [tex]\( 26 \)[/tex], the brother owes Barbara his part of the charge, which is [tex]\( \$41.00 \)[/tex].

Therefore, Barbara's brother owes her:
[tex]\[ \boxed{\$41.00} \][/tex]