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

[prasadini]

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

 

[Prove It]

Re: area

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

Have you learned any calculus yet?

 

[MarkFL]

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?