SOLUTION: I really appreciate the help i'm getting from this website! Question: A rectangle has perimeter P. Find the maximum possible area of the rectangle. Thank you so so so so so so much

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Question 454480: I really appreciate the help i'm getting from this website! Question: A rectangle has perimeter P. Find the maximum possible area of the rectangle. Thank you so so so so so so much for your time and effort!
Found 2 solutions by rfer, solver91311:
Answer by rfer(16322) (Show Source):
You can put this solution on YOUR website!
A square has the maximum area

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Let P represent the fixed perimeter.

Let w represent the width of the rectangle.

Let l represent the length of the rectangle.

The perimeter of a rectangle is:

So

The area of a rectangle is the length times the width so a function for the area in terms of the width is:

Algebra Solution:

The area function is a parabola, opening downward, with vertex at:

Since the parabola opens downward, the vertex represents a maximum value of the area function. The value of the width that gives this maximum value is one-fourth of the given perimeter. Therefore, the shape must be a square, and the area is the width squared.

Calculus Solution:

The area function is continuous and twice differentiable over its domain, therefore there will be a local extrema wherever the first derivative is equal to zero and that extreme point will be a maximum if the second derivative is negative at that point.

Therefore the maximum area is obtained when

And that maximum area is:

John

My calculator said it, I believe it, that settles it