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2. Let's make an open-topped box from an 8.5 x 11-inch sheet of paper by cutting squares of length from cach corner and folding up the sides. Then let's tape the four squares together to make a pencil holder, which is a box without a top or bottom but will still hold pencils if it sits on a desk. See pictures below. 8.5 X X -bottom no bottom 11 (a) What's the maximum combined volume of an open-topped box plus a pencil holder that we can achieve? Hint: What domain makes sense for r in this problem? How do we find a maximum value on such a domain? (b) What do we make if we maximize the combined volume, i.e., what does your solution to (a) result in?

2. Let's make an open-topped box from an 8.5 x 11-inch sheet of paper by cutting squares of length from cach corner and folding up the sides. Then let's tape the four squares together to make a pencil holder, which is a box without a top or bottom but will still hold pencils if it sits on a desk. See pictures below. 8.5 X X -bottom no bottom 11 (a) What's the maximum combined volume of an open-topped box plus a pencil holder that we can achieve? Hint: What domain makes sense for r in this problem? How do we find a maximum value on such a domain? (b) What do we make if we maximize the combined volume, i.e., what does your solution to (a) result in?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Transcription for Educational Website** --- ### Box and Pencil Holder Construction from a Sheet of Paper **Problem Statement:** Let's construct an open-topped box from an 8.5 × 11-inch sheet of paper by cutting squares of length \( x \) from each corner and folding up the sides. Then, tape the four squares together to create a pencil holder, which is a box without a top or bottom but will hold pencils if placed on a desk. See the diagrams below. **Diagrams Explanation:** 1. **First Diagram:** - Represents an 8.5 × 11-inch rectangle. - Squares of length \( x \) are cut from each of the four corners. - Dimensions marked as 8.5 inches and 11 inches, with cuts labeled \( x \). 2. **Second Diagram:** - Illustrates the resulting open-topped box formed by folding up the sides after the squares are cut away. - Labeled as having a "bottom." 3. **Third Diagram:** - Displays the pencil holder, made by taping the removed corner squares together. - Labeled as having "no bottom." **Questions for Exploration:** (a) What’s the maximum *combined* volume of an open-topped box *plus* a pencil holder that we can achieve? *Hint:* What domain makes sense for \( x \) in this problem? How do we find a maximum value on such a domain? (b) What do we make if we maximize the combined volume, i.e., what does your solution to (a) result in? --- This exercise explores optimizing material usage by transforming a single sheet of paper into two functional objects: a box and a pencil holder. Consideration should be given to the domain of \( x \) and the geometric constraints imposed by the dimensions.
Transcribed Image Text:**Transcription for Educational Website** --- ### Box and Pencil Holder Construction from a Sheet of Paper **Problem Statement:** Let's construct an open-topped box from an 8.5 × 11-inch sheet of paper by cutting squares of length \( x \) from each corner and folding up the sides. Then, tape the four squares together to create a pencil holder, which is a box without a top or bottom but will hold pencils if placed on a desk. See the diagrams below. **Diagrams Explanation:** 1. **First Diagram:** - Represents an 8.5 × 11-inch rectangle. - Squares of length \( x \) are cut from each of the four corners. - Dimensions marked as 8.5 inches and 11 inches, with cuts labeled \( x \). 2. **Second Diagram:** - Illustrates the resulting open-topped box formed by folding up the sides after the squares are cut away. - Labeled as having a "bottom." 3. **Third Diagram:** - Displays the pencil holder, made by taping the removed corner squares together. - Labeled as having "no bottom." **Questions for Exploration:** (a) What’s the maximum *combined* volume of an open-topped box *plus* a pencil holder that we can achieve? *Hint:* What domain makes sense for \( x \) in this problem? How do we find a maximum value on such a domain? (b) What do we make if we maximize the combined volume, i.e., what does your solution to (a) result in? --- This exercise explores optimizing material usage by transforming a single sheet of paper into two functional objects: a box and a pencil holder. Consideration should be given to the domain of \( x \) and the geometric constraints imposed by the dimensions.
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