How do I solve these ? ### Educational Website Content#### Problem Descriptiona. Suppose the paper is 11"-wide by 13"-long. Let $ x $ represent the length of the side of the square cutout (in inches), and let $ V $ represent the volume of the box (in cubic inches).i. **Write a formula that expresses $ V $ in terms of $ x $.** \[ V = x(11-2x)(13-2x) \]ii. **If the cutout length increases from 0.5 to 2.6 inches, how much does the volume of the box change by?** - **Change in Volume:** 14.0608 cubic inchesiii. **Estimate the maximum volume for this box?** - *Tip: Use a graphing tool like [www.Desmos.com](http://www.Desmos.com).* - **Maximum Volume:** ___ cubic inchesiv. **What cutout length produces the maximum volume?** - **Cutout Length:** ___ inchesb. Suppose we instead create the box from a 5"-wide by 7"-long sheet of paper.i. **Estimate the maximum volume for this box?** - *Tip: Use a graphing tool like [www.Desmos.com](http://www.Desmos.com).* - **Maximum Volume:** ___ cubic inchesii. **What cutout length produces the maximum volume?** - **Cutout Length:** ___ inches#### Instructions- **Box 1:** Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c. Be sure your variables match those in the question. #### Visual AidThere are green and orange shapes at the top, illustration not provided here. These may assist in visualizing how the box is formed from the piece of paper by cutting and folding along the specified lines.### Diagram/Graph ExplanationThe main aspect of this problem involves visualizing how a piece of paper is transformed into a box. You are given the paper's dimensions and must calculate how the cutout size affects the box's volume, ultimately finding the cutout size that maximizes the volume. Use the graphing tool suggested to plot and find the optimal solutions for the expressions given.

Name: How do I solve these ? ### Educational Website Content#### Problem Des
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Description: How do I solve these ? ### Educational Website Content#### Problem Descriptiona. Suppose the paper is 11"-wide by 13"-long. Let $ x $ represent the length of the side of the square cutout (in inches), and let $ V $ represent the volume of the box

We are entitled to solve only first 3 subparts of the first question so, I am providing the same.…

Math

Advanced Math

a. Suppose the paper is 11"-wide by 13"-long. Let a represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches). i. Write a formula that expresses V in terms of x. V = x(11-2x)(13-2x) ii. If the cutout length increases from 0.5 to 2.6 inches, how much does the volume of the box change by? 14.0608 |1 cubic inches Preview iii. Estimate the maximum volume for this box? Tip: Use a graphing tool like www.Desmos.com. * cubic inches Preview

a. Suppose the paper is 11"-wide by 13"-long. Let a represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches). i. Write a formula that expresses V in terms of x. V = x(11-2x)(13-2x) ii. If the cutout length increases from 0.5 to 2.6 inches, how much does the volume of the box change by? 14.0608 |1 cubic inches Preview iii. Estimate the maximum volume for this box? Tip: Use a graphing tool like www.Desmos.com. * cubic inches Preview

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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Topic Video

Question

How do I solve these ?

### Educational Website Content

#### Problem Description

a. Suppose the paper is 11"-wide by 13"-long. Let $ x $ represent the length of the side of the square cutout (in inches), and let $ V $ represent the volume of the box (in cubic inches).

i. **Write a formula that expresses $ V $ in terms of $ x $.**

\[
V = x(11-2x)(13-2x)
\]

ii. **If the cutout length increases from 0.5 to 2.6 inches, how much does the volume of the box change by?**

- **Change in Volume:** 14.0608 cubic inches

iii. **Estimate the maximum volume for this box?**

- *Tip: Use a graphing tool like [www.Desmos.com](http://www.Desmos.com).*

- **Maximum Volume:** ___ cubic inches

iv. **What cutout length produces the maximum volume?**

- **Cutout Length:** ___ inches

b. Suppose we instead create the box from a 5"-wide by 7"-long sheet of paper.

i. **Estimate the maximum volume for this box?**

- *Tip: Use a graphing tool like [www.Desmos.com](http://www.Desmos.com).*

- **Maximum Volume:** ___ cubic inches

ii. **What cutout length produces the maximum volume?**

- **Cutout Length:** ___ inches

#### Instructions

- **Box 1:** Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c. Be sure your variables match those in the question.

#### Visual Aid

There are green and orange shapes at the top, illustration not provided here. These may assist in visualizing how the box is formed from the piece of paper by cutting and folding along the specified lines.

### Diagram/Graph Explanation

The main aspect of this problem involves visualizing how a piece of paper is transformed into a box. You are given the paper's dimensions and must calculate how the cutout size affects the box's volume, ultimately finding the cutout size that maximizes the volume. Use the graphing tool suggested to plot and find the optimal solutions for the expressions given.

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