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In this video, let's discuss a very interesting property, you might be given an area and you have

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to maintain it constant and you have to find what perimeter satisfies the maximum perimeter.

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So let me try to explain that with an example.

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Say it's given that the area has to be maintained as a 16 centimetre square.

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Now you are free to construct a rectangle with integral sites or the length and breadth of to be integers.

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Now you are asked to find what length and breadth will give you the maximum perimeter.

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So let me summarize.

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You're given the area.

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This area has to be maintained constant and you have to find the length and breadth of the rectangle

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such that the area is equal to 16 centimetres square and the length and breadth have to be integers.

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Now, how do we do this now?

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What are all the different possibilities?

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I could have the length of 16 Berettas one or the length that's eight and Brittas two or the length

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that's four and that's four.

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Right.

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Because in each of these cases, the area is 16, 16 and the one is 16 18.

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The two is 16 four and the four is equal to 16.

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All right.

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Now, is there any other possibility?

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There is no other possibility.

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Right.

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How do we arrive at this?

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Let's quickly see that you have the product of L and B 16 to in what always.

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Can I write this?

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Let me write.

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This has been 12 because generally the bread is the smaller value.

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But it's not wrong if I write it in the other way also.

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So if I take bread one and because the product has to be 16, the length will be 16, they'd run into

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16, gives me 16.

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What if I take the bread as to then the length has to be eight.

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Right.

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Because to do it gives me 16.

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What if I take the industry the length, the bread that's three.

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The length would be 16, divided by three right now, 16 divided by three is not an integer.

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So this is excluded.

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What if the bread is four?

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Then the length also will be sixteen by four.

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That is equal to four.

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What if the bread is five?

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Now, if that is the case, the length has to be sixteen by five.

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And this is excluded because this is not an integer, right?

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You will get a decimal value.

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What if the bread is six?

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Then over here you have sixteen by six.

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That is also excluded.

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What if the bread is seven?

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You have sixteen by seven.

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That is also excluded.

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Right.

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And in this manner, you can see that you only have these three possibilities.

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Now, I will give you a tip.

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Where do we stop to keep going on like this?

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You can stop at this point because you can see that at this point, both of them are equal, because

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after this, the next one that will satisfy this criteria will be eight in do two.

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And you will see that that's already covered over here.

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Right.

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So you can you don't need to double counted it.

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You can stop at the place where either the length and breadth become equal or if they don't become equal,

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they become very close.

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For example, in the case where area is equal to forty eight, you will see that one in the forty eight

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to into twenty four, three in sixteen or in the twelfth.

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Six to eight are all possibilities right now.

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After this you will see that the next one is eighteen to six, which is a repetition.

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So you can stop at this point.

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So either both become equal or they become very close.

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And if the next one is changing, that is it's a repetition over here.

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You can stop at this point.

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Alright, let's proceed.

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We have taken a quick side note to understand how we arrived at these three possibilities, right.

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Let me wrap this piece and make some space over here.

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All right, so these are the three possibilities, that is the length can be 16 and Brett can be one,

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the length can be eight and Brett can be two and the length and before and Brett can be four.

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Now, we have to maximize the perimeter if you have to maximize the perimeter.

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Let's see what we should do.

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Let's find the perimeter.

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And each of these cases over here, you have to in the 16 plus one, that's three, four centimeter.

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Well, here you have two in the eight plus two, that's 20 centimeter.

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And over here you have two in the four plus four, that's 16 centimeter.

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Now you can see that the maximum perimeter is paint over here.

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That is thirty four centimeter.

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And in this case, we have taken the length and breadth as one.

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That is to maximize the some you need to keep the length and breadth as far apart as possible.

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But over here to some because perimeter is two times the length plus Brett.

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Right.

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But to maximize the sum when the product is constant, you need to keep the two numbers as far apart

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as possible.

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But over here we have kept 16 and once we have kept them as far apart as possible.

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Now, let's see this case.

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They are given a constant perimeter, which is at some base number, and you need to maximize the area,

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which is a product based number.

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Right.

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So let's take an example.

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Say the perimeter is equal to 16 centimeter and you have to maximize the area.

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Now, perimeter is equal to 16 means length plus spread is equal to eight.

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Right.

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Because perimeter is two times length plus spread.

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All right.

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Now let's see all the possibilities.

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I could have linked as one and Brett as seven or length as two and Brett six or Lanxess three, Brett

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this five and like this for and Brett has fought beyond that.

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It gets to repeat the next one will be five and three, which is already covered over here.

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So we stop let's find the areas in each of these cases.

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I have seven over here, seven and one.

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Twenty seven.

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Two in the six gives me twelve in this next case, three in the five gives me fifteen and four in to

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four gives me sixteen.

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You can see that the maximum area is obtained over here.

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So to maximize the product that is the area areas product based.

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Right.

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To maximize the product you need to make LMB that is the two numbers as close as possible.

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So let me quickly summarize.

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If you are given the product that is the area is the product and you need to maximize the sum that is

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the perimetral, some based, you need to keep the two numbers as far apart as possible.

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Now, if you are given the sum that is parameter, is sum based and you need to maximize the area,

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the area is product based, you need to keep the two numbers as close as possible.

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Alright, let's do a few practice problems to understand and solidify our learning in this video.