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]
