To determine the monthly bill using the given piecewise function, we need to evaluate the function at [tex]\( x = 800 \)[/tex] kWh. Let's carefully follow the steps laid out by the function [tex]\( b(x) \)[/tex]:
1. First, we identify which part of the piecewise function applies:
- If [tex]\( x \leq 200 \)[/tex], the bill is calculated using [tex]\( 0.10x \)[/tex].
- If [tex]\( x > 200 \)[/tex], the bill is calculated using [tex]\( 0.15(x - 200) + 20 \)[/tex].
Since [tex]\( x = 800 \)[/tex] is greater than 200, we use the second part of the function [tex]\( 0.15(x - 200) + 20 \)[/tex].
2. Substitute [tex]\( x = 800 \)[/tex] into the second part of the function:
[tex]\[
b(800) = 0.15(800 - 200) + 20
\][/tex]
3. Simplify the term inside the parentheses:
[tex]\[
b(800) = 0.15(600) + 20
\][/tex]
4. Perform the multiplication:
[tex]\[
b(800) = 0.15 \times 600 + 20
\][/tex]
[tex]\[
b(800) = 90 + 20
\][/tex]
5. Add the results:
[tex]\[
b(800) = 110
\][/tex]
Thus, the bill for using 800 kWh in a month is [tex]\( \$110 \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{110} \)[/tex].
