SOLUTION: Draw 11 rectangles with a perimeter of 24cm. For the width, use natural numbers, starting at 1. Indicate the length, the width, as well as the area on the sketch of each rectangle

Question 947687: Draw 11 rectangles with a perimeter of 24cm. For the width, use natural numbers, starting at 1. Indicate the length, the width, as well as the area on the sketch of each rectangle
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
This is a misguided, misleading question, but I think I know how the teacher expects you to interpret it.
You can choose for the width (in cm) the natural numbers 1, 2, 3, ....,8, 9, 10, and 11, but
the rectangle with a "width" of will have a "length" of
and will look just the same as
the rectangle with a width of and a length of .
I guess what is meant is for you to draw 11 rectangles,
with a side parallel to left side of the page measuring 1, 2, 3, ... 10, and 11 cm.
We can call the measure of that side the "width".
Then, we call the measure of a side perpendicular to that one
(a side parallel to the top edge of the paper) the "length",
without worrying if the "width" happens to be more than the "length".
Your drawing could look like this
Since , ---> ---> --->
So, the rectangle with a "width" of , will have a length of ;
the rectangle with a "width" of , will have a length of ;
the rectangle with a "width" of , will have a length of ;
the rectangle with a "width" of , will have a length of ;
the rectangle with a "width" of , will have a length of , and so on.
The next (and sixth) "rectangle" has a "width" of , and a "length" of . That is a very special kind of rectangle, with the same measure for width and length. That is the kind of rectangle that we commonly call a square.
For each rectangle you would calculate
.
In square centimeters, the 6 possible areas are:
,
,
,
,
, and
.
What your teacher was trying to make you "discover" is that
for any given perimeter, the largest rectangle is a square.

NOTE: The 7th rectangle would have a with a "width" of , and a "length" of . The measurement that we called "width" is greater than the measurement we call "length". You may think that the 7th rectangle really has a width of and a length of .
You may think that the 7th rectangle really is the same rectangle as the 5th rectangle, just aligned in a different direction. The problem with that line of thought is that it leads you to think that there are only 6 possible rectangles instead of 11.
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