1. ## Rectangle Dimensions

What are the dimensions of a rectangle of a fixed perimeter P that result in the largest area?

NOTE: What exactly is a fixed perimeter?

2. Originally Posted by symmetry
What are the dimensions of a rectangle of a fixed perimeter P that result in the largest area?

NOTE: What exactly is a fixed perimeter?
The dimensions of a rectangle of a fixed perimeter, P, that results in the largest area is a square.

Thus, the dimensions will be P/4 by P/4.

And thus the largest area will be (P/4)^2.

3. ## ok

Okay...but where did you get p/4?

4. Originally Posted by symmetry
Okay...but where did you get p/4?
Every side of a square is equal, and therefore the perimeter of a square is:

x + x + x + x, where x is the length of a side,

Or similarly, 4x = P, where P is the perimeter

P/4 = x, which is a side of the square

Thus, the dimensions that result in the largest area has to be P/4 by P/4.

5. Originally Posted by symmetry
...
NOTE: What exactly is a fixed perimeter?
Hello,

normally the perimeter of a rectangle is calculated by:

$p=2 \cdot length+2 \cdot width$

A fixed perimeter means that the result of this sum is a constant. Length and/or width may change but the perimeter has allways the same value.

EB

6. Originally Posted by symmetry
Okay...but where did you get p/4?
Hello,

to demonstrate what a fixed perimeter means I've attached a digram which shows rectangles which have all the perimeter 16 cm. As you may see the size of those rectangles differs a lot. You'll find the rectangle with the greatest size in the middle of the triangle. I've drawn this square with some thicker perimeter.

Hope this helps a little bit further.

EB

7. ## ok

I thank you for the notes.

Earboth:

You have outdone yourself here. All your replies with diagrams are the best I've seen.