Certainly! Let's go through each part step-by-step to compute the monthly cost for the given anytime minutes used, using the given cost function:
### Given Cost Function
[tex]\[
C(x) = \begin{cases}
19.99 & \text{if } 0 < x \leq 350 \\
0.20x - 50.01 & \text{if } x > 350
\end{cases}
\][/tex]
### Part (a): Compute [tex]\(C(215)\)[/tex]
For [tex]\(x = 215\)[/tex]:
1. Since [tex]\(215\)[/tex] falls in the range [tex]\(0 < x \leq 350\)[/tex],
2. We use the cost function [tex]\(C(x) = 19.99\)[/tex].
Thus, [tex]\(C(215) = 19.99\)[/tex].
### Part (b): Compute [tex]\(C(410)\)[/tex]
For [tex]\(x = 410\)[/tex]:
1. Since [tex]\(410\)[/tex] falls in the range [tex]\(x > 350\)[/tex],
2. We use the cost function [tex]\(C(x) = 0.20x - 50.01\)[/tex].
Substitute [tex]\(x = 410\)[/tex]:
[tex]\[
C(410) = 0.20 \times 410 - 50.01
\][/tex]
[tex]\[
C(410) = 82.00 - 50.01
\][/tex]
[tex]\[
C(410) = 31.99
\][/tex]
Thus, [tex]\(C(410) = 31.99\)[/tex].
### Part (c): Compute [tex]\(C(351)\)[/tex]
For [tex]\(x = 351\)[/tex]:
1. Since [tex]\(351\)[/tex] falls in the range [tex]\(x > 350\)[/tex],
2. We use the cost function [tex]\(C(x) = 0.20x - 50.01\)[/tex].
Substitute [tex]\(x = 351\)[/tex]:
[tex]\[
C(351) = 0.20 \times 351 - 50.01
\][/tex]
[tex]\[
C(351) = 70.20 - 50.01
\][/tex]
[tex]\[
C(351) = 20.19
\][/tex]
Thus, [tex]\(C(351) = 20.19\)[/tex].
### Summary of Results
- [tex]\(C(215) = 19.99\)[/tex]
- [tex]\(C(410) = 31.99\)[/tex]
- [tex]\(C(351) = 20.19\)[/tex]
Each cost is rounded to the nearest cent as specified.
