To calculate the bill for a person who uses 300 kWh in a month using the given piecewise function [tex]\( b(x) \)[/tex], let's walk through the steps.
1. Identify which part of the piecewise function to use:
- The given function is:
[tex]\[
b(x) = \begin{cases}
0.15x & \text{if } x \leq 200 \\
0.10(x - 200) + 30 & \text{if } x > 200
\end{cases}
\][/tex]
- Since [tex]\( x = 300 \)[/tex] kWh, which is greater than 200, we use the second part of the piecewise function: [tex]\( b(x) = 0.10(x - 200) + 30 \)[/tex].
2. Substitute [tex]\( x = 300 \)[/tex] into the appropriate part of the function:
[tex]\[
b(300) = 0.10(300 - 200) + 30
\][/tex]
3. Simplify the expression inside the parentheses first:
[tex]\[
b(300) = 0.10(100) + 30
\][/tex]
4. Perform the multiplication:
[tex]\[
b(300) = 10 + 30
\][/tex]
5. Add the results to find the final bill amount:
[tex]\[
b(300) = 40
\][/tex]
Therefore, the bill for a person who uses 300 kWh in a month is [tex]\( \$40 \)[/tex]. So, the correct answer is:
D. \$40
