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You are constructing a cardboard box from a piece of cardboard with the dimensions 4 m by 8 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions (in m) of the box with the largest volume? 4 m 8 m

You are constructing a cardboard box from a piece of cardboard with the dimensions 4 m by 8 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions (in m) of the box with the largest volume? 4 m 8 m

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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**Title: Optimizing Box Dimensions for Maximum Volume** **Introduction:** In this activity, we will explore how to construct a cardboard box from a flat piece of cardboard, aiming to maximize the box's volume. This problem involves applying concepts of geometry and optimization. **Problem Statement:** You are given a piece of cardboard with dimensions of 4 meters by 8 meters. To form the box, you will cut equal-sized squares from each corner and fold the resulting flaps up to create the sides of the box. Your task is to determine the size of the squares (denoted as 'x') to maximize the volume of the box. **Diagram Explanation:** - The diagram shows a rectangle representing the cardboard. - The cardboard measures 4 meters in width and 8 meters in length. - Four identical squares are cut out from each corner. Each side of these squares is labeled 'x.' - Once cut, these squares will allow you to fold along the dotted lines to form the edges of the box. **Objective:** Determine the value of 'x' that provides the largest possible volume for the box once the sides are folded up. **Key Considerations:** - The length of the box after folding is (8 - 2x) meters. - The width of the box after folding is (4 - 2x) meters. - The height of the box is 'x' meters. The volume 'V' of the box can be expressed as: \[ V = x(8 - 2x)(4 - 2x) \] **Explore this mathematical model to find the optimal value of 'x' that results in the largest box volume.**
Transcribed Image Text:**Title: Optimizing Box Dimensions for Maximum Volume** **Introduction:** In this activity, we will explore how to construct a cardboard box from a flat piece of cardboard, aiming to maximize the box's volume. This problem involves applying concepts of geometry and optimization. **Problem Statement:** You are given a piece of cardboard with dimensions of 4 meters by 8 meters. To form the box, you will cut equal-sized squares from each corner and fold the resulting flaps up to create the sides of the box. Your task is to determine the size of the squares (denoted as 'x') to maximize the volume of the box. **Diagram Explanation:** - The diagram shows a rectangle representing the cardboard. - The cardboard measures 4 meters in width and 8 meters in length. - Four identical squares are cut out from each corner. Each side of these squares is labeled 'x.' - Once cut, these squares will allow you to fold along the dotted lines to form the edges of the box. **Objective:** Determine the value of 'x' that provides the largest possible volume for the box once the sides are folded up. **Key Considerations:** - The length of the box after folding is (8 - 2x) meters. - The width of the box after folding is (4 - 2x) meters. - The height of the box is 'x' meters. The volume 'V' of the box can be expressed as: \[ V = x(8 - 2x)(4 - 2x) \] **Explore this mathematical model to find the optimal value of 'x' that results in the largest box volume.**
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