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What is the maximum possible area of a parallelogram with on

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What is the maximum possible area of a parallelogram with one side of length 2 meters and a perimeter of 24 meters?

Originally posted by jlgdr on 20 Sep 2013, 16:12.
Last edited by Bunuel on 20 Sep 2013, 16:23, edited 1 time in total.
RENAMED THE TOPIC.
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What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 20 Sep 2013, 16:25
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Re: What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 07 May 2015, 00:36
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Bunuel wrote:
jlgdr wrote:
What is the maximum possible area of a parallelogram with one side of length 2 meters and a perimeter of 24 meters?


The perimeter = 2*2 + 2x = 24 --> x = 10.

The are is maximized if the parallelogram is a rectangle, thus the maximum area is 2*10 = 20.


Dear Bunuel:

In this question I have one query that as per the Theory learnt - Max. possible Area with a given perimeter is a SQUARE, so why here it is RECTANGLE?
Pls help with an explanation.
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Re: What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 07 May 2015, 01:04
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anu1706 wrote:
Bunuel wrote:
jlgdr wrote:
What is the maximum possible area of a parallelogram with one side of length 2 meters and a perimeter of 24 meters?


The perimeter = 2*2 + 2x = 24 --> x = 10.

The are is maximized if the parallelogram is a rectangle, thus the maximum area is 2*10 = 20.


Dear Bunuel:

In this question I have one query that as per the Theory learnt - Max. possible Area with a given perimeter is a SQUARE, so why here it is RECTANGLE?
Pls help with an explanation.


Hi anu1706

You are right in your understanding that the maximum possible area for a given perimeter is a square.
So for a give perimeter of 24, the sides should have been 24/4 = 6 units each.

But here we have an additional constraint that the length of one of the sides is 2.
Now, if the other sides are also 2 each, the perimeter will only be 8 and not 24.

So the only way to have a perimeter of 24 and length of one of the sides as 2 is to take the parallelogram as a rectangle, which will result in the answer explained in the posts above.
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Re: What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 07 May 2015, 03:00
Hi anu1706

You are right in your understanding that the maximum possible area for a given perimeter is a square.
So for a give perimeter of 24, the sides should have been 24/4 = 6 units each.

But here we have an additional constraint that the length of one of the sides is 2.
Now, if the other sides are also 2 each, the perimeter will only be 8 and not 24.

So the only way to have a perimeter of 24 and length of one of the sides as 2 is to take the parallelogram as a rectangle, which will result in the answer explained in the posts above.[/quote]

Thanks!! I got it now..
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Re: What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 21 Jul 2018, 02:41
Quote:

Hi anu1706

You are right in your understanding that the maximum possible area for a given perimeter is a square.
So for a give perimeter of 24, the sides should have been 24/4 = 6 units each.

But here we have an additional constraint that the length of one of the sides is 2.
Now, if the other sides are also 2 each, the perimeter will only be 8 and not 24.

So the only way to have a perimeter of 24 and length of one of the sides as 2 is to take the parallelogram as a rectangle, which will result in the answer explained in the posts above.


Thank you so much. Can you give as the proof just to increase our understanding?
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Re: What is the maximum possible area of a parallelogram with on  [#permalink]

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New post 21 Jul 2018, 03:04
Avraheem wrote:
Quote:

Hi anu1706

You are right in your understanding that the maximum possible area for a given perimeter is a square.
So for a give perimeter of 24, the sides should have been 24/4 = 6 units each.

But here we have an additional constraint that the length of one of the sides is 2.
Now, if the other sides are also 2 each, the perimeter will only be 8 and not 24.

So the only way to have a perimeter of 24 and length of one of the sides as 2 is to take the parallelogram as a rectangle, which will result in the answer explained in the posts above.


Thank you so much. Can you give as the proof just to increase our understanding?


I think you're asking why does a rectangle have the maximum area for a given set of sides, am I right?
One way is to see that the area of a parallelogram is ABsinQ (Q being the angle between them), so sin is max when Q=90, hence rectangle.

Another way not involving trigonometry would be, imagine a parallelogram with fixed side lengths and we will be varying angles (base and top = 6, and other sides=2)
For any parallelogram, it is true that area = base * perpendicular height from the base to the top

Now the base length is 6 and fixed, the height is the distance between the top and bottom lines (That is how far the two 6 sides are, perpendicular distance), right?
In which case do you get the max height (distance between the sides)?
When the height is equal to 2, ie when the side of length 2 is actually the height! Again giving you that Q must be 90, hence rectangle.

A way to check this is to take the polar opposite, Q=0/180 degrees, the figure collapses into a line and has area 0.

I hope I was able to get my point straight!
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Re: What is the maximum possible area of a parallelogram with on   [#permalink] 21 Jul 2018, 03:04
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