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Rectangular floors having perimeter of 16 meters are to be covered wit

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Rectangular floors having perimeter of 16 meters are to be covered wit  [#permalink]

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New post 07 Jul 2017, 00:21
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Question Stats:

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Rectangular floors having perimeter of 16 meters are to be covered with carpet squares that measure 1 meter by 1 meter each, costing $6 apiece. What is the maximum possible cost for the number of carpet squares needed to cover any such rectangular floor if the sides of the floors are integers?

(A) $42
(B) $72
(C) $90
(D) $96
(E) $120

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Rectangular floors having perimeter of 16 meters are to be covered wit  [#permalink]

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New post 07 Jul 2017, 03:06
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Bunuel wrote:
Rectangular floors having perimeter of 16 meters are to be covered with carpet squares that measure 1 meter by 1 meter each, costing $6 apiece. What is the maximum possible cost for the number of carpet squares needed to cover any such rectangular floor if the sides of the floors are integers?

(A) $42
(B) $72
(C) $90
(D) $96
(E) $120


If perimeter is 16,

L + W = 8

If we have integer sides, we can have these combinations.

1 ad 7
2 and 6
3 and 5
4 and 4

Since area is maximum when it is a square i.e. 16,

We have \(16 * 6 = 96\)
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Re: Rectangular floors having perimeter of 16 meters are to be covered wit  [#permalink]

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New post 07 Jul 2017, 04:24
You do not have to try any combination of numbers to figure out the maximum. This is because with a 4 sided figure, for any given perimeter, a square maximized the area.

Thus 2x + 4y = 16 becomes 4s = 16;
Each side is 4 so the maximum area is 16;
16*$6 = $96;
Answer D
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Re: Rectangular floors having perimeter of 16 meters are to be covered wit  [#permalink]

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New post 12 Jul 2017, 16:34
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Bunuel wrote:
Rectangular floors having perimeter of 16 meters are to be covered with carpet squares that measure 1 meter by 1 meter each, costing $6 apiece. What is the maximum possible cost for the number of carpet squares needed to cover any such rectangular floor if the sides of the floors are integers?

(A) $42
(B) $72
(C) $90
(D) $96
(E) $120


Since we want the maximum possible cost of the carpet, we want the maximum possible area of a rectangle whose perimeter is 16 meters. The maximum possible area of a rectangle with a given perimeter is a square. Thus, if we let s = side of the square, then 4s = 16 or s = 4 meters. Thus, the area of the square is s^2 = 4^2 = 16 square meters. Since each carpet squares is 1 square meter and costs $6, we need 16 carpet squares totalling 16 x $6 = $96.

Answer: D
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Re: Rectangular floors having perimeter of 16 meters are to be covered wit   [#permalink] 12 Jul 2017, 16:34
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