How to Minimize the Perimeter of a Rectangle Formed by 24 Unit Squares?

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SUMMARY

The smallest rectangle that can be formed using 24 unit squares (1 cm² each) has a perimeter of 20 cm. The perimeter function is defined as P(x,y) = 2(x+y), with the area constraint A = xy. To minimize the perimeter, one can either solve for one variable using the area constraint or apply Lagrange multipliers from calculus. This approach ensures the optimal dimensions of the rectangle are achieved.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly optimization techniques.
  • Familiarity with perimeter and area formulas for rectangles.
  • Knowledge of Lagrange multipliers for constrained optimization.
  • Ability to manipulate algebraic equations to express variables in terms of others.
NEXT STEPS
  • Study optimization techniques in calculus, focusing on minimizing functions.
  • Learn about Lagrange multipliers and their applications in constrained optimization problems.
  • Explore geometric properties of rectangles and their implications on perimeter and area.
  • Practice solving similar optimization problems involving different shapes and constraints.
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Students studying calculus, mathematicians interested in optimization problems, and educators teaching geometric properties and optimization techniques.

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The perimeter of the smallest rectangle that can be formed using 24 squares 1cm2 of area each is
 
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Re: area

prasadini said:
The perimeter of the smallest rectangle that can be formed using 24 squares 1cm2 of area each is

Have you learned any calculus yet?
 
I am assuming the perimeter is the objective function (the function we wish to optimize (minimize in this case)), so let's give that in terms of the width $x$ and height $y$:

$$P(x,y)=2(x+y)$$

Our constraint is on the area $A$, and so we have:

$$A=xy$$

Now, we can solve the constraint for either variable $x$ or $y$, and then write the perimeter function in one variable, and use Calc I techniques of minimization, or we can use the Calc III technique of Lagrange multipliers.

Can you proceed?
 

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