From a 24-inch by 24-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of x. Graph the function and from the graph determine the value of x, to the nearest tenth of an inch, that will yield the maximum volume. OA. 3.8 inches OB. 3.6 inches OC. 4.0 inches OD. 4.2 inches
From a 24-inch by 24-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of x. Graph the function and from the graph determine the value of x, to the nearest tenth of an inch, that will yield the maximum volume. OA. 3.8 inches OB. 3.6 inches OC. 4.0 inches OD. 4.2 inches
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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![### Optimization Problem: Maximizing the Volume of a Box
**Problem Statement:**
From a 24-inch by 24-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let \( x \) represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of \( x \). Graph the function and from the graph determine the value of \( x \), to the nearest tenth of an inch, that will yield the maximum volume.
**Answer Choices:**
- **A.** 3.8 inches
- **B.** 3.6 inches
- **C.** 4.0 inches
- **D.** 4.2 inches
**Explanation:**
To solve the problem, follow these steps:
1. **Understanding the Box Configuration:**
- When squares of side length \( x \) inches are cut from each corner of the 24-inch by 24-inch metal piece, folding the sides up will form a rectangular prism (box).
- The dimensions of the box will be:
- Length: \( 24 - 2x \)
- Width: \( 24 - 2x \)
- Height: \( x \)
2. **Volume of the Box:**
- The volume \( V \) of the rectangular box is calculated by the formula:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the dimensions:
\[
V(x) = (24 - 2x)(24 - 2x)(x)
\]
Simplifying this:
\[
V(x) = x(576 - 96x + 4x^2)
\]
\[
V(x) = 4x^3 - 96x^2 + 576x
\]
3. **Graphing the Function:**
- Graph the function \( V(x) = 4x^3 - 96x^2 + 576x \) to identify the value of \( x \) that maximizes the volume.
- The graph typically shows a peak point which represents the maximum volume.
4. **Determining Maximum Volume:**
- Using calculus or](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e2935af-1329-4dfa-9560-8781a8a0c5be%2F9e15f163-647c-431b-8d87-32cfc5632c5a%2Fwq3w8kl_processed.png&w=3840&q=75)
Transcribed Image Text:### Optimization Problem: Maximizing the Volume of a Box
**Problem Statement:**
From a 24-inch by 24-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let \( x \) represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of \( x \). Graph the function and from the graph determine the value of \( x \), to the nearest tenth of an inch, that will yield the maximum volume.
**Answer Choices:**
- **A.** 3.8 inches
- **B.** 3.6 inches
- **C.** 4.0 inches
- **D.** 4.2 inches
**Explanation:**
To solve the problem, follow these steps:
1. **Understanding the Box Configuration:**
- When squares of side length \( x \) inches are cut from each corner of the 24-inch by 24-inch metal piece, folding the sides up will form a rectangular prism (box).
- The dimensions of the box will be:
- Length: \( 24 - 2x \)
- Width: \( 24 - 2x \)
- Height: \( x \)
2. **Volume of the Box:**
- The volume \( V \) of the rectangular box is calculated by the formula:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the dimensions:
\[
V(x) = (24 - 2x)(24 - 2x)(x)
\]
Simplifying this:
\[
V(x) = x(576 - 96x + 4x^2)
\]
\[
V(x) = 4x^3 - 96x^2 + 576x
\]
3. **Graphing the Function:**
- Graph the function \( V(x) = 4x^3 - 96x^2 + 576x \) to identify the value of \( x \) that maximizes the volume.
- The graph typically shows a peak point which represents the maximum volume.
4. **Determining Maximum Volume:**
- Using calculus or
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