Solve using Lagrange Multipliers **Problem 8**A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in², material for the sides cost 10 cents/in², and the material for the top is 5 cents/in². How do you minimize cost?---This problem involves optimizing the cost of materials for constructing a box with a specific volume constraint. **Key Points for Solution:**1. **Understand the Volume Constraint**: - The box must have a volume of 10,000 cubic inches.2. **Identify Cost Factors**: - Bottom Material: 25 cents per square inch. - Side Material: 10 cents per square inch. - Top Material: 5 cents per square inch.3. **Objective**: - Minimize the total cost while ensuring the volume requirement is met.4. **Approach**: - Set up equations for the volume and cost. - Use calculus or algebraic methods to find dimensions that minimize the total cost.This problem is suitable for illustrating concepts in optimization, especially using calculus to find minimum values given certain constraints.

Answered: Image /qna-images/answer/243c32e6-ebd6-42e1-b33d-3f98de5c5d7c.jpg

Problem 8 A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in?, material for the sides cost 10 cents/in?, and the material for the top is 5 cents/in?. How do you minimize cost?

Problem 8 A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in?, material for the sides cost 10 cents/in?, and the material for the top is 5 cents/in?. How do you minimize cost?

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...

See similar textbooks

Related questions

Q: • Consider an open-Top cardboard in 3-D space with rectangular faces, width x feat breadth y feet…

A: Given: A rectangular box open from the top having rectangular faces is given. The width of the…

Q: A contractor is building a multi-unit commercial building partitioned into five (par- allel)…

A: Given, Building is partitioned into five equally sized units each with area = 300m2 Exterior walls…

Q: A rectangle with side lengths maximum possible value for of x and y has an area of 125 units². What…

A: Given that the sides of the rectangle are x and y and its area is 125 square units

Q: Q4: Design the biggest box (volume) without cover that made from 6m2 of aluminum.

Q: Answer the question below: A gift from an organization is wrapped in a box provided that the sum of…

A: This is the problem of maxima and minima of the function. By using lagrange multipliers we will…

Q: Findthe least squares solutionfor the best parabola going through(2,1), (3,2), (4,2),and(5,4)

A: The given points are (2,1), (3,2), (4,2) and…

Q: A farmer wants to fence an area of 600 m in a rectangular field and then divide it in half with a…

A: Consider the given information,

Q: aving 3m2 of aluminu

A: Given Having 3m2 of aluminum foil at your disposal, you want to build a rectangular box with a lid,…

Q: You are going to fence in three sides of a yard with 400 meters of fence. What will the dimensions…

A: The given yard is in the shape of rectangle and the farmer wants to fence three sides of it, Given…

Q: A farmer has 2400m of fence with which to fence off a rectangular field and divide it into six parts…

A: Consider the figure to cover the complete fencing 4 times y ( that is 4y) and 3 times x ( that is…

Q: Why is the method of Lagrange Multiplies so helpful? Be specific

Q: Suppose you are designing a rectangular room with a fixed perimeter of 60 ft, use the method of…

A: Consider the equation

Q: Why does the process of Lagrange multipliers work for finding extrema?

Q: A closed rectangular box with a volume of 96 cubic meters is to be constructed of two materials. The…

A: Let x be the length y be the width and z be the height of the box Min Z=2xy+2xz+2yz+xy Subjected to…

Q: Find three positive numbers whose product is fixed such that their sum is as small as possible. Use…

A: To write the problem formally, one may find it as, To MinimizeF(x,y,z)=x+y+zsubject…

Q: What are lagrange's multipliess? Beriefly explain. In what conditions lagranges's multiplier method…

A: Lagrangian multiplier is a variable used in the Lagrangian method.

Q: A 100 feet of fencing material is used to build a rectangular fence. (a) area and calculate the…

A: (a) Let the length and breadth of rectangular fence be x and y feet respectively. Perimeter of the…

Question

Solve using Lagrange Multipliers

**Problem 8**

A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in², material for the sides cost 10 cents/in², and the material for the top is 5 cents/in². How do you minimize cost?

---

This problem involves optimizing the cost of materials for constructing a box with a specific volume constraint.

**Key Points for Solution:**

1. **Understand the Volume Constraint**:
- The box must have a volume of 10,000 cubic inches.

2. **Identify Cost Factors**:
- Bottom Material: 25 cents per square inch.
- Side Material: 10 cents per square inch.
- Top Material: 5 cents per square inch.

3. **Objective**:
- Minimize the total cost while ensuring the volume requirement is met.

4. **Approach**:
- Set up equations for the volume and cost.
- Use calculus or algebraic methods to find dimensions that minimize the total cost.

This problem is suitable for illustrating concepts in optimization, especially using calculus to find minimum values given certain constraints.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Pls do fast and i will give like for sure Solution must be in typed form A printed page must contain 80 (cm)^(2) of printed material. There are to be margins of 6 cm on either side and 4 cm on the top and bottom. What are the dimensions of the printed area to minimize the amount of paper to be used?
A rectangular box with no lid will be made of cardboard and have a surface area of 24 square feet. Find the dimensions that will have the maximum possible value. Use Lagrange Multipliers.
pls help asap with solutions
A box without a lid is required to have volume, V = 32000 cm^3. Minimize theamount of cardboard required to create this box. Use the method of Lagrange Multipliers to solve thisproblem. You should get the same answer as you did from HW4. The answer should be 40cm x 40cm x 20cm.
A rectangular play yard is to be constructed along the side of a house by erecting a fence on three sides, using the house wall as the 4th wall of the play yard. Find the dimensions that produce the play yard of maximum area if 20 m of fence is available for the project.
Use method of Lagrange multipliers to find the optimal solution. y= 20A-101.5V phi= A(230-47.7/V) - 5556 =0
Using the Method of Lagrange Multipliers, determine the dimensions of a rectangular box that is open at the top and has a volume of 32 cubic ft, requiring the least amount of material for its construction.