Optimizing Area with Perimeter Constraints

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Homework Help Overview

The problem involves optimizing the area of an equilateral triangle and a square under the constraint that the sum of their perimeters equals 10. Participants are exploring how to express the dimensions of both shapes to minimize the total area.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the formulation of primary and secondary equations related to area and perimeter. There is an attempt to express the height of the triangle in terms of its base, and questions arise about the relationships between the dimensions of the triangle and the square.

Discussion Status

The discussion is active, with participants providing insights on how to relate the dimensions of the triangle and square. Some guidance has been offered regarding the geometric relationships involved, particularly in expressing height in terms of base.

Contextual Notes

Participants are working under the constraint that the sum of the perimeters is fixed at 10, which influences their equations and relationships. There is an acknowledgment of the dependence between the dimensions of the triangle and square.

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Homework Statement


the sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Homework Equations





The Attempt at a Solution


my problem is finding the primary and secondary equations.
is A=1/2bh+s^2 the primary equation and P=3b+4s the secondary equation
 
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Seems ok so far. Now you want to express h in terms of b as well, right? b and h aren't independent.
 
how could i put h in terms of b
first i put b in terms of s and got (10-4s)/3 then put that into the area equation
 
You can put h in terms of b because it's an equilateral triangle. The height is directly proportional to the base. Use trig or pythagoras.
 
thank you very much
 

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