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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#### Given:
- Volume \( V = 40 \) cubic inches.
- Cost for the sides' material: $2 per square inch.
- Cost for the top and bottom material: $10 per square inch.
#### Tasks:
1. **Determine the Material Needed (Surface Area), as a Function of \( x \) and \( y \):**
Material needed =
**(Box Diagram Explanation)**:
The provided diagram depicts a rectangular box with a square base, where \( x \) represents the side length of the base and \( y \) represents the height of the box.
2. **Calculate the Cost of the Material, as a Function of \( x \) and \( y \)**:
\[
C(x, y) =
\]
3. **Express the Cost of the Material as a Function of \( x \) Using the Volume Constraint (Hint: \( V = x^2 y \))**:
\[
C(x) = C(x) =
\]
4. **Differentiate the Cost Function**:
\[
C'(x) =
\]
5. **Solve \( C'(x) = 0 \) to Find the Value of \( x \) That Minimizes the Cost**:
\[
x = \text{inches}
\]
\[
y = \text{inches}
\]
#### Step-by-Step Solutions:
1. **Determine the Material Needed**:
The surface area \( A \) of the box is calculated as follows:
- Four sides: \( 4 \cdot (x \cdot y) \)
- Top and bottom: \( 2 \cdot (x^2) \)
Therefore, the material needed as a function of \( x \) and \( y \) is:
\[
A(x, y) = 4xy + 2x^2
\]
2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dc1f5fb-7f1a-46f5-afa5-8986050fea88%2F03d60782-8fab-4bfa-a481-f4242c4106f0%2Fbno4lll_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Resource on Cost Optimization for a Box with a Square Base
#### Problem Description:
A box, with a square base, is to have a volume of \( V = 40 \) cubic inches. The cost for the materials of the four sides is $2 per square inch, while the cost of the material for the top and bottom is $10 per square inch.
#### Given:
- Volume \( V = 40 \) cubic inches.
- Cost for the sides' material: $2 per square inch.
- Cost for the top and bottom material: $10 per square inch.
#### Tasks:
1. **Determine the Material Needed (Surface Area), as a Function of \( x \) and \( y \):**
Material needed =
**(Box Diagram Explanation)**:
The provided diagram depicts a rectangular box with a square base, where \( x \) represents the side length of the base and \( y \) represents the height of the box.
2. **Calculate the Cost of the Material, as a Function of \( x \) and \( y \)**:
\[
C(x, y) =
\]
3. **Express the Cost of the Material as a Function of \( x \) Using the Volume Constraint (Hint: \( V = x^2 y \))**:
\[
C(x) = C(x) =
\]
4. **Differentiate the Cost Function**:
\[
C'(x) =
\]
5. **Solve \( C'(x) = 0 \) to Find the Value of \( x \) That Minimizes the Cost**:
\[
x = \text{inches}
\]
\[
y = \text{inches}
\]
#### Step-by-Step Solutions:
1. **Determine the Material Needed**:
The surface area \( A \) of the box is calculated as follows:
- Four sides: \( 4 \cdot (x \cdot y) \)
- Top and bottom: \( 2 \cdot (x^2) \)
Therefore, the material needed as a function of \( x \) and \( y \) is:
\[
A(x, y) = 4xy + 2x^2
\]
2.
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