Trail Rev... = 360 cubic inches. olData Career... A box, with a square base is to have a volume of V The cost for the materials of the four sides is $6 per square inch, while the cost of the material for the top and bottom is $10 per square inch. y X X Determine the material needed (surface area), as a function of x and y. Material = The cost of the material, as a function of x and y is C = The cost of the material, as a function of x is C = C(x) = (Hint: V = y) Sign
Trail Rev... = 360 cubic inches. olData Career... A box, with a square base is to have a volume of V The cost for the materials of the four sides is $6 per square inch, while the cost of the material for the top and bottom is $10 per square inch. y X X Determine the material needed (surface area), as a function of x and y. Material = The cost of the material, as a function of x and y is C = The cost of the material, as a function of x is C = C(x) = (Hint: V = y) Sign
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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![### Optimizing the Cost of a Box with a Square Base
#### Problem Statement
A box, with a square base, is to have a volume of \( V = 360 \) cubic inches. The cost for the materials of the four sides is $6 per square inch, while the cost of the material for the top and bottom is $10 per square inch.
#### Diagram Explanation
The diagram provided is a three-dimensional representation of the box with the following dimensions indicated:
- The base dimensions are \( x \) and \( x \) (since it is a square base).
- The height of the box is \( y \).
#### Mathematical Formulation
1. **Volume Constraint:**
The volume \( V \) of the box is given by:
\[ V = x^2 y = 360 \, \text{cubic inches} \]
2. **Surface Area Calculation:**
To determine the material needed, calculate the surface area:
- **Four sides:**
Each side has an area of \( x \cdot y \). With four sides, the total area is:
\[ 4 \cdot (x \cdot y) \]
- **Top and Bottom:**
Each has an area of \( x \cdot x = x^2 \). With two faces (top and bottom), the total area is:
\[ 2 \cdot x^2 \]
Summing these up, the total surface area \( A \) is:
\[ A = 4xy + 2x^2 \]
3. **Cost Calculation:**
The cost for the materials, represented as a function of \( x \) and \( y \) is calculated by incorporating the respective costs for each surface area:
- **Cost for the four sides:**
\[ \text{Cost}_{\text{four sides}} = 6 \cdot (4xy) \]
- **Cost for the top and bottom:**
\[ \text{Cost}_{\text{top and bottom}} = 10 \cdot (2x^2) \]
Adding these together, the total cost \( C \), as a function of \( x \) and \( y \), is:
\[ C = 6 \cdot 4xy + 10 \cdot 2x^2 \]
Simplified, this becomes:
\[ C = 24xy + 20x^2 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf909276-6ad3-4144-b98c-235c5a32e436%2F893001be-2704-4154-8122-1ac6a517b1a5%2Fsh8lcha_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Optimizing the Cost of a Box with a Square Base
#### Problem Statement
A box, with a square base, is to have a volume of \( V = 360 \) cubic inches. The cost for the materials of the four sides is $6 per square inch, while the cost of the material for the top and bottom is $10 per square inch.
#### Diagram Explanation
The diagram provided is a three-dimensional representation of the box with the following dimensions indicated:
- The base dimensions are \( x \) and \( x \) (since it is a square base).
- The height of the box is \( y \).
#### Mathematical Formulation
1. **Volume Constraint:**
The volume \( V \) of the box is given by:
\[ V = x^2 y = 360 \, \text{cubic inches} \]
2. **Surface Area Calculation:**
To determine the material needed, calculate the surface area:
- **Four sides:**
Each side has an area of \( x \cdot y \). With four sides, the total area is:
\[ 4 \cdot (x \cdot y) \]
- **Top and Bottom:**
Each has an area of \( x \cdot x = x^2 \). With two faces (top and bottom), the total area is:
\[ 2 \cdot x^2 \]
Summing these up, the total surface area \( A \) is:
\[ A = 4xy + 2x^2 \]
3. **Cost Calculation:**
The cost for the materials, represented as a function of \( x \) and \( y \) is calculated by incorporating the respective costs for each surface area:
- **Cost for the four sides:**
\[ \text{Cost}_{\text{four sides}} = 6 \cdot (4xy) \]
- **Cost for the top and bottom:**
\[ \text{Cost}_{\text{top and bottom}} = 10 \cdot (2x^2) \]
Adding these together, the total cost \( C \), as a function of \( x \) and \( y \), is:
\[ C = 6 \cdot 4xy + 10 \cdot 2x^2 \]
Simplified, this becomes:
\[ C = 24xy + 20x^2 \]
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