piecewise function based on a real-life situation.like selling shoes

A piecewise function can be very useful for modeling real-life situations where different conditions lead to different outcomes. Let's take an example of selling shoes with a discount system based on the quantity sold.

### Real-Life Scenario:
A shoe store offers the following pricing scheme:
- For the first 10 pairs of shoes, the price per pair is $50.
- For any shoes beyond the first 10 pairs, there is a 20% discount, reducing the price per pair to $40.

We want to create a piecewise function to represent the total cost based on the number of pairs of shoes purchased, $ n $.

### Piecewise Function:
\[
C(n) = \begin{cases} 
50n & \text{if } 0 \leq n \leq 10, \\
500 + 40(n - 10) & \text{if } n > 10.
\end{cases}
\]

#### Explanation:
1. **When $ 0 \leq n \leq 10 $**:
   - If you buy 10 or fewer pairs, each pair costs $50. So, the total cost is $ 50n $.
   
2. **When $ n > 10 $**:
   - For the first 10 pairs, you pay $500 (since $ 50 \times 10 = 500 $).
   - For any additional pairs beyond the first 10, you pay $40 per pair. Therefore, the total cost for $ n > 10 $ is $500 plus $40 for each pair after 10, which is $ 40(n - 10) $.

### Example Calculation:
- If someone buys 8 pairs, the total cost would be:
  \[
  C(8) = 50 \times 8 = 400.
  \]
  
- If someone buys 15 pairs, the total cost would be:
  \[
  C(15) = 500 + 40(15 - 10) = 500 + 40 \times 5 = 500 + 200 = 700.
  \]

Would you like further details or clarifications on this function?

Here are 5 related questions to deepen your understanding:
1. What would the cost be for buying exactly 10 pairs of shoes?
2. How would the total cost change if the discount started after 5 pairs instead of 10?
3. Can you model a scenario where different price tiers are applied, such as three different price categories?
4. How can we modify this function if the first few pairs of shoes are given for free?
5. What would happen if the discount percentage changed to 30% instead of 20%?

**Tip:** Piecewise functions are great tools for modeling situations with different conditions, such as thresholds or limits, often found in pricing, taxes, and other financial calculations.

piecewise function based on a real-life situation. like selling shoes

Explore how to use piecewise functions to model real-life scenarios like selling shoes with a discount system based on quantity sold. This problem illustrates key algebraic concepts, including piecewise functions and linear equations, suitable for high school students in Grades 9-12.

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