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
Please solve Q6
![### Problem Q6: Maximizing the Volume of a Box
#### Problem Statement:
A box is shown in the figure below. Find \( x \) so that the volume is maximized.
#### Diagram Description:
The problem presents a two-dimensional figure of a rectangular sheet measuring 5 feet by 8 feet. The corners of the rectangle are to be cut out to form a box. Each corner cut-out is a smaller square with side length \( x \). After cutting out these squares, the sides of the remaining figure are folded up along the dotted lines to form the box.
On the left side:
- The original dimensions of the rectangle are labeled as 5 feet in height and 8 feet in width.
- Square cut-outs from each corner are marked with side length \( x \).
On the right side:
- Illustrates how the remaining shape would be folded along the dotted lines (labeled as "Fold lines") to form an open-top box.
#### Steps to Solve:
1. **Calculate the New Dimensions After Cutting:**
- The length of the new box will be \( (8 - 2x) \) feet.
- The width of the new box will be \( (5 - 2x) \) feet.
- The height of the new box will be \( x \) feet.
2. **Volume of the Box:**
- The volume \( V \) of the newly formed box can be expressed as:
\[
V(x) = (8 - 2x)(5 - 2x)x
\]
3. **Maximizing the Volume:**
- Determine the value of \( x \) that maximizes the volume \( V \). This can be done by finding the critical points of the function \( V(x) \) and evaluating them.
4. **Constraints:**
- \( x \) must be such that the dimensions make sense (i.e., \( x \) should be positive and less than half the length and width of the original rectangle, so \( 0 < x < 2.5 \)).
By calculating the above steps, you can find the value of \( x \) that maximizes the volume of the box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cc04b1e-803c-4d9d-b550-d2ae04bcc2b3%2F7f4ddb7c-fdc0-495d-a617-c1d4654579eb%2Fh7nuulk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Q6: Maximizing the Volume of a Box
#### Problem Statement:
A box is shown in the figure below. Find \( x \) so that the volume is maximized.
#### Diagram Description:
The problem presents a two-dimensional figure of a rectangular sheet measuring 5 feet by 8 feet. The corners of the rectangle are to be cut out to form a box. Each corner cut-out is a smaller square with side length \( x \). After cutting out these squares, the sides of the remaining figure are folded up along the dotted lines to form the box.
On the left side:
- The original dimensions of the rectangle are labeled as 5 feet in height and 8 feet in width.
- Square cut-outs from each corner are marked with side length \( x \).
On the right side:
- Illustrates how the remaining shape would be folded along the dotted lines (labeled as "Fold lines") to form an open-top box.
#### Steps to Solve:
1. **Calculate the New Dimensions After Cutting:**
- The length of the new box will be \( (8 - 2x) \) feet.
- The width of the new box will be \( (5 - 2x) \) feet.
- The height of the new box will be \( x \) feet.
2. **Volume of the Box:**
- The volume \( V \) of the newly formed box can be expressed as:
\[
V(x) = (8 - 2x)(5 - 2x)x
\]
3. **Maximizing the Volume:**
- Determine the value of \( x \) that maximizes the volume \( V \). This can be done by finding the critical points of the function \( V(x) \) and evaluating them.
4. **Constraints:**
- \( x \) must be such that the dimensions make sense (i.e., \( x \) should be positive and less than half the length and width of the original rectangle, so \( 0 < x < 2.5 \)).
By calculating the above steps, you can find the value of \( x \) that maximizes the volume of the box.
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