A rectangular box with open top and has a square base is a school project of two senior high school students. However, they are wondering what would be the minimum dimensions of their box if it will have a fixed volume of 62.5 cm³. Steps 1. Draw a diagram. List what is asked on the problem and label the diagram with relevant data. 2. Write the constraint and the optimization equations. 3. Substitute the constraint equations, to the corresponding length, width and height of the optimization equation. 4. Simplify and take its first derivative. 5. Set the equation to zero and solve for the x value (critical point). 6. If there are two critical values, we have to check which one will give a sensible answer by substituting them to the volume equation. 7. Test the x value. Substitute it to the second derivative and check whether the answer is less than or greater than zero. 8. Substitute the maximum x value to the simplified constraint equation to solve for I and w. 9. Solve for the dimensions. Solution Constraint equation: Optimization equation: height= length width=
A rectangular box with open top and has a square base is a school project of two senior high school students. However, they are wondering what would be the minimum dimensions of their box if it will have a fixed volume of 62.5 cm³. Steps 1. Draw a diagram. List what is asked on the problem and label the diagram with relevant data. 2. Write the constraint and the optimization equations. 3. Substitute the constraint equations, to the corresponding length, width and height of the optimization equation. 4. Simplify and take its first derivative. 5. Set the equation to zero and solve for the x value (critical point). 6. If there are two critical values, we have to check which one will give a sensible answer by substituting them to the volume equation. 7. Test the x value. Substitute it to the second derivative and check whether the answer is less than or greater than zero. 8. Substitute the maximum x value to the simplified constraint equation to solve for I and w. 9. Solve for the dimensions. Solution Constraint equation: Optimization equation: height= length width=
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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Answer the Problems, Show your Complete Solutions (This is all about Calculus: Optimization Problems).
I already provided the answers I just need the COMPLETE SOLUTIONS.
a. Length = width = 5 m, Height = 2.5 m
b. x = 135 m, y = 67.5 m
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