HARRELL RENTZ
  • HOME
  • IDEAS
    • MINDFUL MOMENTS
    • SEE-THINK-WONDERS
    • QFTs
    • ACTIVITIES >
      • DESMOS
      • MATHEMATICA
      • PROJECTS
      • GEOGEBRA
      • KHAN ACADEMY
      • VIDEOS
      • MISCELLANEOUS
    • RUBRICS
  • COURSES
    • AP CALCULUS AB >
      • L01
      • L02
      • L03
      • L04
      • D01 old
      • D02
      • D03
      • D04
      • D05
      • D06
      • D07
      • D08 >
        • Max Volume Sample
      • N01
      • N02
      • N03
      • N04
      • N07
      • N08
      • N09
      • AP EXAM PREP
      • P01
      • P02
      • P03
      • P04
      • P05
      • P06
      • P07
      • P08
      • P09
      • P10
    • ALGEBRA 2 >
      • Linear Regression
  • MY STORY
    • THEATRE-LOVER
  • GRATITUDE

Optimization Sample Solution: Maximizing the Volume of a Rectangular Solid Made from a Sheet of Paper

The Problem

You have an 8.5" x 11" (US Letter Size) sheet of paper.
You want to make an open-top paper box by cutting a square out of each corner and then folding up the sides.
What is the length of the side of each square (corner) cut-out that will produce the maximum possible volume for the box?

1. Create a function model

Picture
Let x represent the side of the square cut-outs from the corners.
Write a Volume function: V(x)

Picture

2. Figure out the domain of your function: what values might x have?

The smallest value that x might have is zero. The associated volume would be 0.
The largest value that x might have is (1/2)(8.5) or 4.25. (This is half the length of the shortest side.
​So, the practical domain for our function is [0,4.25]. The associated volume here would also be 0.

3. "Optimize" the model, i.e. find the x value that produces the global maximum value for function V(x). Use our procedure for finding extrema!

We know this will occur at a critical point (a place where the derivative is 0 or undefined) or an endpoint.
The derivative is never 0 for this polynomial function.
Where is the derivative equal to 0?
Picture
The table of important values to consider would be as follows:
​
Picture

4. If graphing is an option, a TI calculator or Desmos graph can help you approximate this solution as follows:
     Desmos Graph

Proudly powered by Weebly
  • HOME
  • IDEAS
    • MINDFUL MOMENTS
    • SEE-THINK-WONDERS
    • QFTs
    • ACTIVITIES >
      • DESMOS
      • MATHEMATICA
      • PROJECTS
      • GEOGEBRA
      • KHAN ACADEMY
      • VIDEOS
      • MISCELLANEOUS
    • RUBRICS
  • COURSES
    • AP CALCULUS AB >
      • L01
      • L02
      • L03
      • L04
      • D01 old
      • D02
      • D03
      • D04
      • D05
      • D06
      • D07
      • D08 >
        • Max Volume Sample
      • N01
      • N02
      • N03
      • N04
      • N07
      • N08
      • N09
      • AP EXAM PREP
      • P01
      • P02
      • P03
      • P04
      • P05
      • P06
      • P07
      • P08
      • P09
      • P10
    • ALGEBRA 2 >
      • Linear Regression
  • MY STORY
    • THEATRE-LOVER
  • GRATITUDE