In this video, let's discuss a very interesting property, you might be given an area and you have
to maintain it constant and you have to find what perimeter satisfies the maximum perimeter.
So let me try to explain that with an example.
Say it's given that the area has to be maintained as a 16 centimetre square.
Now you are free to construct a rectangle with integral sites or the length and breadth of to be integers.
Now you are asked to find what length and breadth will give you the maximum perimeter.
So let me summarize.
You're given the area.
This area has to be maintained constant and you have to find the length and breadth of the rectangle
such that the area is equal to 16 centimetres square and the length and breadth have to be integers.
Now, how do we do this now?
What are all the different possibilities?
I could have the length of 16 Berettas one or the length that's eight and Brittas two or the length
that's four and that's four.
Right.
Because in each of these cases, the area is 16, 16 and the one is 16 18.
The two is 16 four and the four is equal to 16.
All right.
Now, is there any other possibility?
There is no other possibility.
Right.
How do we arrive at this?
Let's quickly see that you have the product of L and B 16 to in what always.
Can I write this?
Let me write.
This has been 12 because generally the bread is the smaller value.
But it's not wrong if I write it in the other way also.
So if I take bread one and because the product has to be 16, the length will be 16, they'd run into
16, gives me 16.
What if I take the bread as to then the length has to be eight.
Right.
Because to do it gives me 16.
What if I take the industry the length, the bread that's three.
The length would be 16, divided by three right now, 16 divided by three is not an integer.
So this is excluded.
What if the bread is four?
Then the length also will be sixteen by four.
That is equal to four.
What if the bread is five?
Now, if that is the case, the length has to be sixteen by five.
And this is excluded because this is not an integer, right?
You will get a decimal value.
What if the bread is six?
Then over here you have sixteen by six.
That is also excluded.
What if the bread is seven?
You have sixteen by seven.
That is also excluded.
Right.
And in this manner, you can see that you only have these three possibilities.
Now, I will give you a tip.
Where do we stop to keep going on like this?
You can stop at this point because you can see that at this point, both of them are equal, because
after this, the next one that will satisfy this criteria will be eight in do two.
And you will see that that's already covered over here.
Right.
So you can you don't need to double counted it.
You can stop at the place where either the length and breadth become equal or if they don't become equal,
they become very close.
For example, in the case where area is equal to forty eight, you will see that one in the forty eight
to into twenty four, three in sixteen or in the twelfth.
Six to eight are all possibilities right now.
After this you will see that the next one is eighteen to six, which is a repetition.
So you can stop at this point.
So either both become equal or they become very close.
And if the next one is changing, that is it's a repetition over here.
You can stop at this point.
Alright, let's proceed.
We have taken a quick side note to understand how we arrived at these three possibilities, right.
Let me wrap this piece and make some space over here.
All right, so these are the three possibilities, that is the length can be 16 and Brett can be one,
the length can be eight and Brett can be two and the length and before and Brett can be four.
Now, we have to maximize the perimeter if you have to maximize the perimeter.
Let's see what we should do.
Let's find the perimeter.
And each of these cases over here, you have to in the 16 plus one, that's three, four centimeter.
Well, here you have two in the eight plus two, that's 20 centimeter.
And over here you have two in the four plus four, that's 16 centimeter.
Now you can see that the maximum perimeter is paint over here.
That is thirty four centimeter.
And in this case, we have taken the length and breadth as one.
That is to maximize the some you need to keep the length and breadth as far apart as possible.
But over here to some because perimeter is two times the length plus Brett.
Right.
But to maximize the sum when the product is constant, you need to keep the two numbers as far apart
as possible.
But over here we have kept 16 and once we have kept them as far apart as possible.
Now, let's see this case.
They are given a constant perimeter, which is at some base number, and you need to maximize the area,
which is a product based number.
Right.
So let's take an example.
Say the perimeter is equal to 16 centimeter and you have to maximize the area.
Now, perimeter is equal to 16 means length plus spread is equal to eight.
Right.
Because perimeter is two times length plus spread.
All right.
Now let's see all the possibilities.
I could have linked as one and Brett as seven or length as two and Brett six or Lanxess three, Brett
this five and like this for and Brett has fought beyond that.
It gets to repeat the next one will be five and three, which is already covered over here.
So we stop let's find the areas in each of these cases.
I have seven over here, seven and one.
Twenty seven.
Two in the six gives me twelve in this next case, three in the five gives me fifteen and four in to
four gives me sixteen.
You can see that the maximum area is obtained over here.
So to maximize the product that is the area areas product based.
Right.
To maximize the product you need to make LMB that is the two numbers as close as possible.
So let me quickly summarize.
If you are given the product that is the area is the product and you need to maximize the sum that is
the perimetral, some based, you need to keep the two numbers as far apart as possible.
Now, if you are given the sum that is parameter, is sum based and you need to maximize the area,
the area is product based, you need to keep the two numbers as close as possible.
Alright, let's do a few practice problems to understand and solidify our learning in this video.