Question 9 A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 450 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.05 cents per square centimeter. Find the dimensions for the package that will minimize production cost. Helpful information: h : height of cylinder, r: radius of cylinder Volume of a cylinder: V = ²h Area of the sides: A = 2πrh Area of the top/bottom: A = ² To minimize the cost of the package: Radius: Height: Minimum cost: cm cm cents
Question 9 A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 450 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.05 cents per square centimeter. Find the dimensions for the package that will minimize production cost. Helpful information: h : height of cylinder, r: radius of cylinder Volume of a cylinder: V = ²h Area of the sides: A = 2πrh Area of the top/bottom: A = ² To minimize the cost of the package: Radius: Height: Minimum cost: cm cm cents
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Optimization Problem: Designing a Cost-Effective Soup Container
A microwaveable cup-of-soup package needs to be designed in the shape of a cylinder to hold 450 cubic centimeters of soup. The goal is to find the dimensions that minimize production costs. The sides and bottom of the container will be made from styrofoam, costing 0.04 cents per square centimeter, while the top will be made of glued paper, costing 0.05 cents per square centimeter.
---
#### Helpful Information
- **\( h \)**: height of the cylinder
- **\( r \)**: radius of the cylinder
#### Formulas
1. **Volume of a Cylinder**:
\[
V = \pi r^2 h
\]
2. **Area of the Sides**:
\[
A = 2 \pi r h
\]
3. **Area of the Top/Bottom**:
\[
A = \pi r^2
\]
---
#### Objective
Minimize the production cost of the package by determining the optimal radius and height of the cylinder.
- **Radius**: \( \_\_\_\_\_ \) cm
- **Height**: \( \_\_\_\_\_ \) cm
- **Minimum Cost**: \( \_\_\_\_\_ \) cents
---
By using the given formulas, you can apply calculus techniques to find the values of \( r \) and \( h \) that minimize the total cost while maintaining the required volume of 450 cubic centimeters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc29c810b-c4cc-47ca-9579-623151f0bc7a%2F1afb7374-e3c8-4227-a4da-f75bcff85418%2Fjdjcn8g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Optimization Problem: Designing a Cost-Effective Soup Container
A microwaveable cup-of-soup package needs to be designed in the shape of a cylinder to hold 450 cubic centimeters of soup. The goal is to find the dimensions that minimize production costs. The sides and bottom of the container will be made from styrofoam, costing 0.04 cents per square centimeter, while the top will be made of glued paper, costing 0.05 cents per square centimeter.
---
#### Helpful Information
- **\( h \)**: height of the cylinder
- **\( r \)**: radius of the cylinder
#### Formulas
1. **Volume of a Cylinder**:
\[
V = \pi r^2 h
\]
2. **Area of the Sides**:
\[
A = 2 \pi r h
\]
3. **Area of the Top/Bottom**:
\[
A = \pi r^2
\]
---
#### Objective
Minimize the production cost of the package by determining the optimal radius and height of the cylinder.
- **Radius**: \( \_\_\_\_\_ \) cm
- **Height**: \( \_\_\_\_\_ \) cm
- **Minimum Cost**: \( \_\_\_\_\_ \) cents
---
By using the given formulas, you can apply calculus techniques to find the values of \( r \) and \( h \) that minimize the total cost while maintaining the required volume of 450 cubic centimeters.
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