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The problem you're dealing with is a piecewise function that describes the cost of shipping a package based on its weight $ x $ in pounds. The shipping company, IPS, has different rates for different weight ranges, and the goal is to construct the cost function $ C(x) $.

### Given Information:
1. For packages **weighing less than or equal to 5 pounds**, the rate is **$1.25 per pound**.
2. For packages **weighing between 5 and 13 pounds**, the rate is **$2 per pound**.
3. For packages **weighing 13 pounds or more**, the rate is **$3 per pound**, and an **additional flat fee of $5** applies.
4. **No packages** are accepted if they weigh **more than 80 pounds**.
5. An additional **$1.50 flat fee** applies to **all deliveries**.

### To Create the Piecewise Function:

You want to write a function $ C(x) $ that represents the cost of shipping, where $ x $ is the weight of the package in pounds.

The general structure of the piecewise function is:

\[
C(x) = 
\begin{cases} 
1.25x + 1.5 & \text{if } 0 < x \leq 5 \\
2x + 1.5 & \text{if } 5 < x < 13 \\
3x + 5 + 1.5 & \text{if } 13 \leq x \leq 80
\end{cases}
\]

- For the first interval $ 0 < x \leq 5 $, the cost is $ 1.25x $ plus the $1.50 flat fee.
- For the second interval $ 5 < x < 13 $, the cost is $ 2x $ plus the $1.50 flat fee.
- For the third interval $ 13 \leq x \leq 80 $, the cost is $ 3x $, plus the flat fee of $5 and an additional $1.50.

### Answering the Question:

#### "How much does it cost to ship a package weighing 6 pounds with IPS?"

For a package weighing 6 pounds, it falls in the interval $ 5 < x < 13 $, which means the cost is given by the second case:

\[
C(6) = 2(6) + 1.5 = 12 + 1.5 = 13.5
\]

Thus, the cost to ship a 6-pound package is **$13.50**.

Would you like further details or have any questions? 

Here are 5 related questions you could explore:
1. What would the cost be for a package weighing exactly 5 pounds?
2. What is the cost of shipping a package that weighs 13 pounds?
3. How does the flat fee affect the total cost for a package in each weight range?
4. How would the function change if IPS removed the $1.50 flat fee for all deliveries?
5. What is the maximum cost to ship a package weighing 80 pounds?

**Tip**: For piecewise functions, always carefully define the intervals based on the given conditions to avoid errors in the ranges.

How much does it cost to ship a package weighing 6 pounds with IPS? Create a piecewise function that describes the shipping costs in dollars as a function of the weight in pounds.

Learn how to create a piecewise function to describe the shipping rates for IPS based on package weight. Includes detailed steps to construct the function and solve for the cost of shipping a 6-pound package.

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