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# Marie wishes to enclose a rectangular region in her backyard using

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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
question: if the house is 50 foot long, why can't the opposite side also be 50 foot long?

gmatophobia wrote:
houston1980 wrote:
Marie wished to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?

(A) 10
(B) 20
(C) 25
(D) 40
(E) 50

We can use the options to work backward.

Let AB denote Marie's house.

Given

AB = 50 feet

2x + y = 80

Each option denotes the possible value of y. Using that value, we can find the value of x , and obtain the area.

The working is shown in the image below.

Option D
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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
JeffTargetTestPrep wrote:
houston1980 wrote:
Marie wished to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?

(A) 10
(B) 20
(C) 25
(D) 40
(E) 50

If the side that is bounded by the house has a length of x, then the opposite side is also x, and each of the adjacent sides has a length of (80 – x)/2.

The area of the rectangular region is:

x(80 – x)/2 = (-1/2)x^2 + 40x

The quadratic expression above takes its maximum value if:

x = -b/(2a) = -40/[2(-1/2)] = -40/(-1) = 40

Jeff, isn't square a rectangle with biggest area? Shouldnt all sides be equal?
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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
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Vaishvii wrote:
JeffTargetTestPrep wrote:
houston1980 wrote:
Marie wished to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?

(A) 10
(B) 20
(C) 25
(D) 40
(E) 50

If the side that is bounded by the house has a length of x, then the opposite side is also x, and each of the adjacent sides has a length of (80 – x)/2.

The area of the rectangular region is:

x(80 – x)/2 = (-1/2)x^2 + 40x

The quadratic expression above takes its maximum value if:

x = -b/(2a) = -40/[2(-1/2)] = -40/(-1) = 40

Jeff, isn't square a rectangle with biggest area? Shouldnt all sides be equal?

Vaishvii you are correct that if there is a fixed total for the perimeter, then a square produces the maximum area. This thought was my first instinct on this problem too, but we have to read carefully:

houston1980 wrote:
using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides.

So, the fixed total of 80 feet of fencing is only for "the other 3 sides". And the 4th side can be up to 50 feet.

Once we realize this, I find that the quickest approach is to work backwards from the answer choices to find the maximum, as gmatophobia showed above, and I've repasted it below.

My number sense instinct said that the answer was most likely 40, since we essentially get that wall "for free", without having to use any of the 80 feet of fencing. The problem with the answer "50" is that it is too long and narrow and therefore produces a smaller area — this is the concept you were talking about.

(Optional timesaver: we don't need to test answers A and B — once we see that C is lower than D, we know that these more extreme values will be lower than C, because the graph of a quadratic equation is a "parabola." When there's a maximum, it's an upside-down U-shape, pasted below for your reference.)
Attachments

2023-11-17 16_34_20-Marie wishes to enclose a rectangular region in her backyard using _ Problem Sol.png [ 351.02 KiB | Viewed 6743 times ]

Area vs Width Parabola Graph.png [ 111.21 KiB | Viewed 6736 times ]

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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
Thank you Ben, this is helpful!

GMATCoachBen wrote:
Vaishvii wrote:
houston1980 wrote:
Marie wished to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?

(A) 10
(B) 20
(C) 25
(D) 40
(E) 50

If the side that is bounded by the house has a length of x, then the opposite side is also x, and each of the adjacent sides has a length of (80 – x)/2.

The area of the rectangular region is:

x(80 – x)/2 = (-1/2)x^2 + 40x

The quadratic expression above takes its maximum value if:

x = -b/(2a) = -40/[2(-1/2)] = -40/(-1) = 40

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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
Can questions be asked from Geometry in the gmat focus edition ? Bunuel bb MartyMurray
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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]

sayan640 wrote:
Can questions be asked from Geometry in the gmat focus edition ? Bunuel bb MartyMurray

­Specific knowledge of geometry is not tested in the GMAT Focus Edition. Some Coordinate Geometry topics, such as lines and functions, are still included. However, knowing the area of a rectangle, which is simply length × width, is not considered geometry knowledge by GMAT. It's regarded as general knowledge. This question, for example, tests algebra, not geometry.
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Re: Marie wishes to enclose a rectangular region in her backyard using [#permalink]
houston1980 wrote:
Marie wishes to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?

(A) 10
(B) 20
(C) 25
(D) 40
(E) 50

­
Another approach:
Say the length of the house that is a part of the rectangle is L.
We are given that L + 2B = 80

We know that for positive numbers, given a constant sum, the product is maximum when the numbers are equal.

L/2 + L/2 + B + B = 80
$$\frac{L^2*B^2}{4}$$ is maximum (and hence LB is maximum) when L/2 = B

Hence, $$L + 2* \frac{L}{2} = 80$$ which gives L = 40