9814

Tin Box with Maximum Volume

Problem: A piece of sheet tin three feet square is to be made into a rectangular box open at the top by cutting out equal squares from the corners and bending up the sides of the resulting piece parallel with the edges. Among all such boxes, to find the box of greatest volume.
This is the problem J. L. Walsh used in his 1947 Classroom Note in The American Mathematical Monthly to illustrate a rigorous analysis of maximum-minimum problems. A version of the problem appears in many calculus books and in Walsh’s 1962 booklet.
Let the tin sheet have dimensions , with , and suppose a square with side is cut from each corner. The volume of the resulting box is , .
Walsh’s rigorous analysis uses the extreme value theorem: A continuous function on a closed bounded interval has minimum and maximum values, and the critical point theorem: If a function has an extreme value at an interior point of an interval, its derivative at the point is either zero or does not exist. A proof of the extreme value theorem is best left to an advanced calculus course, but the critical point theorem depends only on the definition of derivative. An extreme value cannot occur where or .
Since is continuous on , it has a maximum value there, by the extreme value theorem.
Let the maximum occur at .
Since , and when , we have .
Since exists for , the critical point theorem implies .
Since , for when .
Therefore, with , the box has maximum volume when .
Calculations are easiest and the biggest box is easiest to make when and are integers and is rational. These are the triples with this property, where and and are relatively prime:
This Demonstration lets you choose one of these triples and an integer multiplier. For example, choosing the first triple and multiplier 3 gives , the case in the stated problem. It is somewhat surprising how shallow the biggest box is.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Former Harvard Professor Joseph Leonard Walsh expanded on his 1947 Classroom Note in a 1962 booklet [2] concerned with rigor in finding global maximum and minimum values. He argued the second derivative is unnecessary.
[1] J. L. Walsh, "A Rigorous Treatment of the First Maximum Problem in the Calculus," The American Mathematical Monthly, 54(1), 1947 pp. 35–36.
[2] J. L. Walsh, A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, Boston: Heath, 1962.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.







Related Curriculum Standards

US Common Core State Standards, Mathematics



 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+