We are constructing a box from a piece of paper. The paper is a piece of "ledger-sized paper" which measures 11"×17". We will remove a square of size "x" inches from each corner and turn up the edges. A piece of ledger paper is the same size as taping the long sides of two pieces of standard 8.5×11 paper together.**Diagram Explanation:**1. **Initial Paper Diagram:** - A rectangle measuring 11" in height and 17" in width. - Squares labeled "x" are shown at each corner of the rectangle indicating the portions to be removed.2. **3D Box Diagram:** - After removing the squares from each corner and turning up the edges, a 3D box shape is depicted.**Instructions:**Once we remove the squares of size "x" inches from each corner and turn up the edges, we create a box. Label the dimensions of the newly created box using the variable "x".**Equations:**- $ h = $- $ w = $The diagrams include empty input boxes under each label where the respective formulas or dimensions for height $ h $ and width $ w $ are to be entered, using mathematical symbols provided above the input areas. ### Volume of a Box: Analyzing the Function of Cut Size#### Problem IntroductionWe are tasked with determining the volume of a box as a function of the cut size, $ x $, from a flat sheet.#### Volume Function Equation- **Equation for Volume $ V(x) $:** - Input the equation in the provided equation box to express the volume $ V(x) $ as a function of $ x $.#### Restricted Domain- **Determine the restricted domain of this problem:** - Identify the range of x values that make sense for the problem. - Input the numbers for the minimum and maximum acceptable values of $ x $.#### Restricted Range- **Determine the restricted range of this problem:** - Identify the possible volume values $ V(x) $ that make sense. - Ensure the range is specific and rounded to two decimal places as required.#### Maximization of Volume- **Maximize the box volume:** - Calculate the cut size $ x $ that maximizes the box's volume. - Write down the optimal $ x $ value in inches.- **Maximum Volume:** - Determine and state the maximum volume $ V $ in cubic inches.#### Largest Cut Size for Specific Volume- **Calculate the largest cut size for a desired volume:** - Find the largest $ x $ that still results in a box volume of $ 130 \, \text{in}^3 $.Use this framework as a guide to solve the problem by filling in the appropriate numbers and equations in each part.

According to guidelines I'm solving 3 subparts only. Please post other subparts as a separate…

What is the equation that represents the Volume of the box as a function of the cutsize of the box? V (x) = ab P a 브 √a |a| k sin (a) E What is the restricted domain of this problem? (That is, what x values "make sense"?) Number ≤x≤Number

What is the equation that represents the Volume of the box as a function of the cutsize of the box? V (x) = ab P a 브 √a |a| k sin (a) E What is the restricted domain of this problem? (That is, what x values "make sense"?) Number ≤x≤Number

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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Question

We are constructing a box from a piece of paper. The paper is a piece of "ledger-sized paper" which measures 11"×17". We will remove a square of size "x" inches from each corner and turn up the edges.

A piece of ledger paper is the same size as taping the long sides of two pieces of standard 8.5×11 paper together.

**Diagram Explanation:**

1. **Initial Paper Diagram:**
- A rectangle measuring 11" in height and 17" in width.
- Squares labeled "x" are shown at each corner of the rectangle indicating the portions to be removed.

2. **3D Box Diagram:**
- After removing the squares from each corner and turning up the edges, a 3D box shape is depicted.

**Instructions:**
Once we remove the squares of size "x" inches from each corner and turn up the edges, we create a box. Label the dimensions of the newly created box using the variable "x".

**Equations:**

- $ h = $

- $ w = $

The diagrams include empty input boxes under each label where the respective formulas or dimensions for height $ h $ and width $ w $ are to be entered, using mathematical symbols provided above the input areas.

### Volume of a Box: Analyzing the Function of Cut Size

#### Problem Introduction
We are tasked with determining the volume of a box as a function of the cut size, $ x $, from a flat sheet.

#### Volume Function Equation
- **Equation for Volume $ V(x) $:**
- Input the equation in the provided equation box to express the volume $ V(x) $ as a function of $ x $.

#### Restricted Domain
- **Determine the restricted domain of this problem:**
- Identify the range of x values that make sense for the problem.
- Input the numbers for the minimum and maximum acceptable values of $ x $.

#### Restricted Range
- **Determine the restricted range of this problem:**
- Identify the possible volume values $ V(x) $ that make sense.
- Ensure the range is specific and rounded to two decimal places as required.

#### Maximization of Volume
- **Maximize the box volume:**
- Calculate the cut size $ x $ that maximizes the box's volume.
- Write down the optimal $ x $ value in inches.

- **Maximum Volume:**
- Determine and state the maximum volume $ V $ in cubic inches.

#### Largest Cut Size for Specific Volume
- **Calculate the largest cut size for a desired volume:**
- Find the largest $ x $ that still results in a box volume of $ 130 \, \text{in}^3 $.

Use this framework as a guide to solve the problem by filling in the appropriate numbers and equations in each part.

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