2. An open box is to be made from a square piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 in. by 6 in.
2. An open box is to be made from a square piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 in. by 6 in.
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...
Related questions
Question
![**Problem 2:**
An open box is to be made from a square piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 inches by 6 inches.
*Explanation:*
- **Objective:** To determine the size of the squares to cut from each corner to maximize the volume of the resulting box.
- **Given:** A 6-inch by 6-inch square piece of material.
- **Process:** Calculate the volume of the box in terms of the size of the cut squares and maximize this volume.
**Steps for Solution:**
1. **Define Variables:** Let \( x \) be the side length of the square cut from each corner.
2. **Volume Equation:** Volume \( V \) of the box is given by:
\[
V = x(6 - 2x)(6 - 2x)
\]
3. **Maximize Volume:** Find \( x \) that maximizes \( V \).
**Study Tip:**
- Apply calculus techniques such as differentiation to find maximum points.
- Remember to consider the domain of \( x \), since \( x \) must be positive and \( 2x \leq 6 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c34ee-e27f-46b4-a2d2-ab5f4ef78243%2F0bf213e3-5489-46fc-92dc-9a3abc8501d2%2Fbqvju3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2:**
An open box is to be made from a square piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 inches by 6 inches.
*Explanation:*
- **Objective:** To determine the size of the squares to cut from each corner to maximize the volume of the resulting box.
- **Given:** A 6-inch by 6-inch square piece of material.
- **Process:** Calculate the volume of the box in terms of the size of the cut squares and maximize this volume.
**Steps for Solution:**
1. **Define Variables:** Let \( x \) be the side length of the square cut from each corner.
2. **Volume Equation:** Volume \( V \) of the box is given by:
\[
V = x(6 - 2x)(6 - 2x)
\]
3. **Maximize Volume:** Find \( x \) that maximizes \( V \).
**Study Tip:**
- Apply calculus techniques such as differentiation to find maximum points.
- Remember to consider the domain of \( x \), since \( x \) must be positive and \( 2x \leq 6 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning