To determine the bill for a person who uses [tex]\(560\, \text{kWh}\)[/tex] in a month using the piecewise function [tex]\( b(x) \)[/tex], we'll follow these steps:
1. Identify which part of the piecewise function we need to use. The given function is:
[tex]\[
b(x)=\begin{cases}
0.15x & \text{if } x \leq 360 \\
0.10(x-360) + 54 & \text{if } x > 360
\end{cases}
\][/tex]
Here, [tex]\(x = 560\)[/tex], which is greater than [tex]\(360\)[/tex]. Therefore, we use the second part of the piecewise function, [tex]\( b(x) = 0.10(x-360) + 54 \)[/tex].
2. Substitute [tex]\( x = 560 \)[/tex] into the second part of the function:
[tex]\[
b(560) = 0.10(560 - 360) + 54
\][/tex]
3. Calculate the expression inside the parentheses:
[tex]\[
560 - 360 = 200
\][/tex]
4. Multiply the result by [tex]\(0.10\)[/tex]:
[tex]\[
0.10 \times 200 = 20
\][/tex]
5. Add the constant term [tex]\(54\)[/tex] to the result:
[tex]\[
20 + 54 = 74
\][/tex]
Therefore, the bill for a person who uses [tex]\(560\, \text{kWh}\)[/tex] in a month is:
[tex]\[ \boxed{74} \][/tex]
