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The perimeter of a rectangular yard is completely surrounded by a fenc

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The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 21 Mar 2016, 07:42
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The perimeter of a rectangular yard is completely surrounded by a fence that measures 40 meters. What is the length of the yard if the area of the yard is 64 meters squared?

A. 8
B. 10
C. 12
D. 14
E. 16

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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 22 Mar 2016, 00:47
Perimeter of rectangular yard = 2(l + b) = 40 --> l + b = 20

Area = l * b = 64

b = 20 - l
l(20 - l) = 64
20l - l^2 = 64
l^2 - 20l + 64 = 0

Upon simplifying we get l = 16 or 4. Only 16 is there in the answer choice.

Answer: E
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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 19 Aug 2016, 10:17
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Bunuel wrote:
The perimeter of a rectangular yard is completely surrounded by a fence that measures 40 meters. What is the length of the yard if the area of the yard is 64 meters squared?

A. 8
B. 10
C. 12
D. 14
E. 16

Image
Based on the figure, we dont need to solve the equation, just substitute the values.
If we consider A) 8, as L the W will be 12, which is C, and there can't be 2 right answers, so A and C are out.
And E alone will satisfy both the equation 16 + 4 = 20 and 16 * 4 = 64




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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 19 Aug 2016, 10:44
1
As all answer choices are integers I can find the factors of 64 and see if the sum of a factor pair makes 40. That would be my answer.

64=2^6
1 64
2 32
4 16
8 8

The only one that satisfies the requirements is 4 and 16. As the length is the longest side then 16 is our answer.

Answer choice: E.
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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 31 Dec 2016, 03:21
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Let us solve this problem via back solving method.


Let’s start with C, the middle value. If the length of the yard is 12 and the perimeter is 40, the width would be 8 (perimeter – 2l = 2w). With a length of 12 and a width of 8, the area would be 96. This is too big of an area.

It may not be intuitive whether we need the length to be longer or shorter, based on the above outcome. Consider the following geometric principle: for a fixed perimeter, the maximum area will be achieved when the values for the length and width are closest to one another. A (10 × 10) rectangle has a much bigger area than an 18 × 2 rectangle. Put differently, when dealing with a fixed perimeter, the greater the disparity between the length and the width, the smaller the area.

Since we need the area to be smaller than 96, it makes sense to choose a longer length so that the disparity between the length and width will be greater.

When we get to answer choice E, we see that a length of 16 gives us a width of 4 (perimeter – 2l = 2w). Now the area is in fact 16 × 4 = 64.

The correct answer is E
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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 22 Jan 2018, 16:41
Hi All,

This question can also be solved by TESTing THE ANSWERS.

We're told that the Perimeter of the yard is 40 and its Area is 64. We're asked for the Length of the yard.

Perimeter = 2L + 2W = 40
Area = (L)(W)

Since the answer choices are all integers, my suspicion is that both the length and width of the yard are integers. Which provides an interesting opportunity, since 64 can be "broken down" into just a few options:

1x64
2x32
4x16
8x8

From the given answers, my guess would be either A or E would be the answer. I'll test those options to see if either fits the info in the prompt.

If the yard was 8x8, then the perimeter would be 32, which is NOT a match (the perimeter is supposed to be 40).
If the yard was 4x16, then the perimeter would be 40, which IS a match.

So, either 4 or 16 could be the solution.

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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 22 Jan 2018, 17:03
Bunuel wrote:
The perimeter of a rectangular yard is completely surrounded by a fence that measures 40 meters. What is the length of the yard if the area of the yard is 64 meters squared?

A. 8
B. 10
C. 12
D. 14
E. 16


Given, \((l+b) = 20, l*b = 64\).. Now, as per the answer options l is an integer... so no need of substituting and making it a quadratic equation.. the only integer to satisfy the two will be \(l = 16.\)
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Re: The perimeter of a rectangular yard is completely surrounded by a fenc [#permalink]

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New post 19 Apr 2018, 14:15
Top Contributor
Bunuel wrote:
The perimeter of a rectangular yard is completely surrounded by a fence that measures 40 meters. What is the length of the yard if the area of the yard is 64 meters squared?

A. 8
B. 10
C. 12
D. 14
E. 16


Let L = the length of the rectangle
Let W = the width of the rectangle

The perimeter is 40 meters.
So, L + L + W + W = 40
Simplify: 2L + 2W = 40
Divide both sides by 2 to get: L + W = 20

The area is 64 square meters.
So, LW = 64

We now have two equations.

First, if L + W = 20, then W = 20 - L

Now take LW = 64 and replace W with (20 - L) to get: L(20 - L) = 64
Expand to get: 20L - L² = 64
Rearrange to get: L² - 20L + 64 = 0
Factor: (L - 4)(L - 16) = 0
So, L = 4 or L = 16

So, it LOOKS like we have two different solutions. However, they are actually the SAME solution.

If L = 4, then W = 16
If L = 16, then W = 4

So, the dimensions of the rectangle are 4 X 16. So, we can say that the length is EITHER 4 or 16.

Since only 16 appears among the answer choices, the correct answer must be E

Cheers,
Brent
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Re: The perimeter of a rectangular yard is completely surrounded by a fenc   [#permalink] 19 Apr 2018, 14:15
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