4. In a certain year, a postal service would deliver a first-class letter weighing 3.5 ounces or less for the following prices: not more than 1 ounce, 46¢; more than 1 ounce but not more than 2 ounces, 73¢; more than 2 ounces but not more than 3 ounces, 91¢; and more than 3 ounces but not more than 3.5 ounces, $1.13. Use this information to answer the following. a. Write a piecewise function, P(w), that determines the price P in cents as a func- tion of the weight w in ounces. b. Sketch the graph of P(w). c. Use the information from (a) and (b) to determine the values in the open interval (0, 3.5) that the function is discontinuous.
4. In a certain year, a postal service would deliver a first-class letter weighing 3.5 ounces or less for the following prices: not more than 1 ounce, 46¢; more than 1 ounce but not more than 2 ounces, 73¢; more than 2 ounces but not more than 3 ounces, 91¢; and more than 3 ounces but not more than 3.5 ounces, $1.13. Use this information to answer the following. a. Write a piecewise function, P(w), that determines the price P in cents as a func- tion of the weight w in ounces. b. Sketch the graph of P(w). c. Use the information from (a) and (b) to determine the values in the open interval (0, 3.5) that the function is discontinuous.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Postal Service Pricing Problem
#### Problem Statement
In a certain year, a postal service would deliver a first-class letter weighing 3.5 ounces or less for the following prices:
- Not more than 1 ounce, 46¢;
- More than 1 ounce but not more than 2 ounces, 73¢;
- More than 2 ounces but not more than 3 ounces, 91¢;
- More than 3 ounces but not more than 3.5 ounces, $1.13.
Use this information to answer the following:
#### Part (a)
**a. Write a piecewise function, \( P(w) \), that determines the price \( P \) in cents as a function of the weight \( w \) in ounces.**
The piecewise function \( P(w) \) can be represented as follows:
\[
P(w) =
\begin{cases}
46 & \text{if } 0 < w \leq 1 \\
73 & \text{if } 1 < w \leq 2 \\
91 & \text{if } 2 < w \leq 3 \\
113 & \text{if } 3 < w \leq 3.5
\end{cases}
\]
#### Part (b)
**b. Sketch the graph of \( P(w) \).**
The graph of \( P(w) \) can be described as a step function where each interval of \( w \) corresponds to a different constant price. Below is a detailed description of the graph:
- The graph starts at \( (0, 0) \) and remains at \( 46 \) cents for \( 0 < w \leq 1 \).
- It jumps to \( 73 \) cents at \( w = 1 \) and stays constant for \( 1 < w \leq 2 \).
- At \( w = 2 \), the graph jumps to \( 91 \) cents and stays constant for \( 2 < w \leq 3 \).
- Finally, the graph jumps to \( 113 \) cents at \( w = 3 \) and remains for \( 3 < w \leq 3.5 \).
#### Graph Explanation
The \( x \)-axis represents the weight of the letter \( w \) in ounces, while the \( y \)-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0cd55c-6f84-412a-ac81-9a79eaea810f%2F65a4e5d5-e254-4505-be92-61a166d0eb90%2Fu92890nk_processed.png&w=3840&q=75)
Transcribed Image Text:### Postal Service Pricing Problem
#### Problem Statement
In a certain year, a postal service would deliver a first-class letter weighing 3.5 ounces or less for the following prices:
- Not more than 1 ounce, 46¢;
- More than 1 ounce but not more than 2 ounces, 73¢;
- More than 2 ounces but not more than 3 ounces, 91¢;
- More than 3 ounces but not more than 3.5 ounces, $1.13.
Use this information to answer the following:
#### Part (a)
**a. Write a piecewise function, \( P(w) \), that determines the price \( P \) in cents as a function of the weight \( w \) in ounces.**
The piecewise function \( P(w) \) can be represented as follows:
\[
P(w) =
\begin{cases}
46 & \text{if } 0 < w \leq 1 \\
73 & \text{if } 1 < w \leq 2 \\
91 & \text{if } 2 < w \leq 3 \\
113 & \text{if } 3 < w \leq 3.5
\end{cases}
\]
#### Part (b)
**b. Sketch the graph of \( P(w) \).**
The graph of \( P(w) \) can be described as a step function where each interval of \( w \) corresponds to a different constant price. Below is a detailed description of the graph:
- The graph starts at \( (0, 0) \) and remains at \( 46 \) cents for \( 0 < w \leq 1 \).
- It jumps to \( 73 \) cents at \( w = 1 \) and stays constant for \( 1 < w \leq 2 \).
- At \( w = 2 \), the graph jumps to \( 91 \) cents and stays constant for \( 2 < w \leq 3 \).
- Finally, the graph jumps to \( 113 \) cents at \( w = 3 \) and remains for \( 3 < w \leq 3.5 \).
#### Graph Explanation
The \( x \)-axis represents the weight of the letter \( w \) in ounces, while the \( y \)-
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