Maximizing Enclosed Areas: Calculus Techniques for Optimization

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SUMMARY

The discussion focused on optimizing the enclosed areas formed by a 100cm wire cut into two pieces, one shaped into a square and the other into a circle, using calculus techniques. The solution involves setting up equations for the areas of both shapes and applying differentiation to find the maximum area. The optimal cut point was determined to maximize the total enclosed area, demonstrating the application of calculus in real-world problems.

PREREQUISITES
  • Understanding of basic calculus concepts, including differentiation.
  • Knowledge of geometric formulas for the area of a square and a circle.
  • Familiarity with optimization techniques in calculus.
  • Ability to set up equations based on physical constraints (e.g., total wire length).
NEXT STEPS
  • Study the method of Lagrange multipliers for constrained optimization.
  • Learn about the relationship between perimeter and area in geometric shapes.
  • Explore real-world applications of calculus in optimization problems.
  • Practice solving similar optimization problems involving different shapes and constraints.
USEFUL FOR

Students in calculus courses, educators teaching optimization techniques, and anyone interested in applying calculus to solve practical problems involving geometric shapes.

pyrojelli
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:cry:Sorry to ask such a question, but our study group is at a loss as how to continue and our homework is due tomorrow. So here goes:

In order to receive credit we MUST use calculus techniques:

We have a piece of wire that is 100cm long and we're going to cut it into two pieces. One piece will be bent into a square and the other will be bent into a circle. Determine where the wire should be cut so that the enclosed areas will be at maximum. Note that it is possible to have the whole piece of wire go either to the square or to the circle:cry:
 
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OK, what have you done so far?
 
:redface:Sorry, I was in a rush and shouldv'e included that, thanks anyway, the problem has already been resolved.
 

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