Does a Rectangle's Area Always Increase with Its Perimeter?

[circa415]

If the perimeter of a rectangle increases, does the area necessarily increase?

Can anyone explain this using calculus?

 

[arildno]

No it does not, and you don't need calculus to find the answer:
Let's say you've got a square initially with side "a".
Then, you look at the degenerate rectangle with two sides 3a, and the other two length 0.

The perimeter of the degenerate rectangle is 3a+0+3a+0=6a, that is, greater than your original square's 4a, yet the rectangle's area is zero..

 

[shmoe]

Think of a rectangle with corners at (0,0), (0,1/x), (x,1/x), (x,0). Draw a picture-one corner at the origin the opposite on the graph 1/x.

Now we get an entire family of rectangles parameterized by x>0. What can you say about the area of these guys? What about the perimiter as x varies?