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The perimeter of a rectangle ABCD is 20. What is the maximum area of

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Bunuel
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The perimeter of a rectangle ABCD is 20. What is the maximum area of [#permalink] New post   04 Nov 2021, 05:45
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The perimeter of a rectangle ABCD is 20. What is the maximum area of rectangle ABCD?

(A) 1
(B) 20
(C) 25
(D) 50
(E) 100
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C
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The perimeter of a rectangle ABCD is 20. What is the maximum area of [#permalink] New post   04 Nov 2021, 06:07
Option (C): 25

Given, Perimeter= 20.
Let the length and width be, \(l \) and \(w \) respectively.

So, \(2l + 2w = 20.\)
This gives, \(l + w = 10\).
Since, l and w are positive no.s, they are less than \(10\).

Possible combinations of l and w are:
(\(1,9\)) = This gives area = \(9 \) units.
(\(2,8\)) = This gives area = \(16 \) units.
(\(3,7\)) = This gives area = \(21 \)units.
(\(4,6\)) = This gives area = \(24 \) units.
(\(5,5\)) = This gives area = \(25 \) units.

As you can see, the max area is when the sides are (5,5). Thus, option (C) is the correct answer.
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Re: The perimeter of a rectangle ABCD is 20. What is the maximum area of [#permalink] New post   04 Nov 2021, 07:05
perimeter of rectangle = 2(L+B)
20 = 2(L+B)
thus, L+B = 10
when can the area be maximum??
when L=B , which is equal to 5
therefore, maximum area of a rectangle = 25
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Re: The perimeter of a rectangle ABCD is 20. What is the maximum area of [#permalink] New post   04 Nov 2021, 12:06
General Concept: Given a set sum of values, you can maximize the product of those values by making them equal or as close to equal as possible


In geometry: given a set perimeter of constant length, you can maximize the area of the 2-D polygon by making it a Regular Polygon

Perimeter of the Rectangle = 2L + 2W = 20

The AREA (the Product of the Values L * W) will be MAXIMIZED when the values are EQUAL (L = W) ———

The area of the rectangle will be maximized, given the set perimeter of 20, when the rectangle is a Regular Polygon ——> in this case a SQUARE where L = W

2L + 2W = 20

2L + 2L = 20

4L = 20

L = 5

Area = (L)^2 = (5)^2 = 25


25

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Re: The perimeter of a rectangle ABCD is 20. What is the maximum area of [#permalink]
04 Nov 2021, 12:06
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