Math problem illustrating a piecewise function related to Lucy's monthly sales bonus structure.

Problem Analysis

The problem involves creating a piecewise function to model Lucy's bonus based on the number of sales she makes in a month. The conditions are as follows:
1. If Lucy makes 20 or fewer sales, she earns a bonus of \$25 per sale.
2. If Lucy makes more than 20 sales, she earns \$25 for each of the first 20 sales and an additional \$40 for each sale beyond 20.

We need to:
- Part A: Write a piecewise function \( B(s) \) that models Lucy's bonus for making \( s \) sales.
- Part B: Calculate Lucy's bonus for August when she made 18 sales.
- Part C: Calculate Lucy's bonus for September when she made 25 sales.

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Part A: Writing the Piecewise Function

To write the piecewise function \( B(s) \), we need to consider the two cases described in the problem:

1. Case 1: If \( s \leq 20 \) (Lucy makes 20 or fewer sales).
- Lucy earns \$25 for each sale.
- Therefore, her bonus is \( B(s) = 25s \).

2. Case 2: If \( s > 20 \) (Lucy makes more than 20 sales).
- Lucy earns \$25 for each of the first 20 sales, which totals \( 25 \times 20 = 500 \).
- For each sale beyond 20, she earns an additional \$40.
- The number of sales beyond 20 is \( s - 20 \).
- Therefore, her bonus is \( B(s) = 500 + 40(s - 20) \).

Combining these two cases, the piecewise function is:
\[
B(s) =
\begin{cases}
25s & \text{if } s \leq 20 \\
500 + 40(s - 20) & \text{if } s > 20
\end{cases}
\]

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Part B: Calculating Lucy's Bonus for August

In August, Lucy made 18 sales. Since \( s = 18 \leq 20 \), we use the first case of the piecewise function:
\[
B(s) = 25s
\]
Substitute \( s = 18 \):
\[
B(18) = 25 \times 18 = 450
\]

Thus, Lucy's bonus for August is:
\[
\boxed{450}
\]

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Part C: Calculating Lucy's Bonus for September

In September, Lucy made 25 sales. Since \( s = 25 > 20 \), we use the second case of the piecewise function:
\[
B(s) = 500 + 40(s - 20)
\]
Substitute \( s = 25 \):
\[
B(25) = 500 + 40(25 - 20)
\]
Simplify the expression:
\[
B(25) = 500 + 40 \times 5 = 500 + 200 = 700
\]

Thus, Lucy's bonus for September is:
\[
\boxed{700}
\]

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Final Answers

- Part A: The piecewise function is:
\[
B(s) =
\begin{cases}
25s & \text{if } s \leq 20 \\
500 + 40(s - 20) & \text{if } s > 20
\end{cases}
\]
- Part B: Lucy's bonus for August is \(\boxed{450}\).
- Part C: Lucy's bonus for September is \(\boxed{700}\).