https://file.aiquickdraw.com/mathbot/file/1726202916483jjrvv2cy.jpg

The problem involves defining a piecewise function for shipping costs based on the weight of a parcel, and it is divided into three distinct intervals with corresponding rates. Let's break it down:

1. **For $ x \in [0, 5] $:**
   The rate is $1 per pound, so the cost function for this range is:
   \[
   C(x) = 1 \cdot x
   \]

2. **For $ x \in (5, 13] $:**
   The rate is $2.25 per pound, so the cost function for this range is:
   \[
   C(x) = 2.25 \cdot x
   \]

3. **For $ x \in (13, 83] $:**
   The rate is $3 per pound, so the cost function for this range is:
   \[
   C(x) = 3 \cdot x
   \]

Additionally, there is a flat fee of $2.50 for all deliveries and an additional $5 fee for deliveries weighing more than 13 pounds.

Now, let’s define the piecewise function incorporating all of these elements:

\[
C(x) =
\begin{cases}
1 \cdot x + 2.50 & \text{if } x \in [0, 5] \\
2.25 \cdot x + 2.50 & \text{if } x \in (5, 13] \\
3 \cdot x + 7.50 & \text{if } x \in (13, 83]
\end{cases}
\]

### Answer for the Package Weighing 63 Pounds:
For $ x = 63 $ pounds (which falls in the interval $ (13, 83] $), the cost is:

\[
C(63) = 3 \cdot 63 + 7.50 = 189 + 7.50 = 196.50
\]

So, the cost to ship a package weighing 63 pounds is **$196.50**.

Would you like further clarification on any of the steps? Here are five related questions:

1. How would the cost function change if the weight limit was extended beyond 83 pounds?
2. Can you explain why the function is piecewise and not linear across all weights?
3. What would the cost be for a package weighing exactly 5 pounds?
4. How does the inclusion of a flat fee impact the structure of the piecewise function?
5. If the additional fee for heavier packages increased, how would this affect the cost function?

**Tip:** When working with piecewise functions, always check which interval your input falls into before applying the corresponding formula.

Shipping company IPS has shipping rates that are a function of the weight of the parcel. For packages weighing less than or equal to 5 pounds, they charge $1 per pound. For packages weighing between 5 and 13 pounds, they charge $2.25 per pound. For packages weighing 13 pounds or more, they charge $3 per pound. IPS refuses to ship any parcel weighing more than 83 pounds. There is also a flat fee of $2.50 for all deliveries and an additional fee of $5 for all deliveries weighing 13 pounds or more. Complete the following description of a piecewise function so that it describes the shipping costs C(x) in dollars as a function of the weight x in pounds.

Learn how to calculate shipping costs using a piecewise function for IPS, based on parcel weight. The piecewise function handles different rates for weights between 0-83 pounds, including flat fees for parcels over 13 pounds. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation