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A rectangular yard (white area) is surrounded by a hedge (shaded regio

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A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 13 Nov 2014, 09:37
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Tough and Tricky questions: Geometry.



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Note: Figure not drawn to scale.

A rectangular yard (white area) is surrounded by a hedge (shaded region) with a length of 15 meters and a width of 12 meters, as shown in the figure above. The area of the yard is equal to the area of the hedge. If the ratio of the length of the hedge to the width of the hedge is the same as the ratio of the length of the yard to the width of the yard, what is the width, in meters, of the yard?


A. \(6\sqrt{2}\)

B. \(\frac{5}{4}\)

C. \(\frac{6}{\sqrt{2}}\)

D. \(7\frac{\sqrt{2}}{2}\)

E. \(\frac{6}{2}\)

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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 13 Nov 2014, 19:26
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Answer = A \(= 6\sqrt{2}\)

Total Area = 15*12 = 180

Given that Area of rectangular yard = Area of Hedge (shaded region) \(= \frac{180}{2} = 90\)

Lets say width of the yard = a

then length of the yard \(= \frac{90}{a}\)

Equating the ratios:

\(\frac{15}{12} = \frac{\frac{90}{a}}{a}\)

\(a^2 = 72\)

\(a = 6\sqrt{2}\)
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 13 Nov 2014, 12:39
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Given a ratio of 5:4 I used length of 10 and width of 8 for the yard as a starting point. The area of the yard would then be 10x8=80m and the area of the hedge would be (15x12)-80 = 180-80=100. This tells us that with length 10 and width 8, the yard is not as big as the hedge. Therefore, the width must be larger than 8.

A. knowing that the square root of 2 is roughly 1.4, this answer is approx. 8.4. this is what we are looking for but we need to check to see if any other choices are larger than 8
B. too small
C. too small
D. too small
E. too small

Answer A!
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 13 Nov 2014, 15:41
DangerPenguin wrote:
Given a ratio of 5:4 I used length of 10 and width of 8 for the yard as a starting point. The area of the yard would then be 10x8=80m and the area of the hedge would be (15x12)-80 = 180-80=100. This tells us that with length 10 and width 8, the yard is not as big as the hedge. Therefore, the width must be larger than 8.

A. knowing that the square root of 2 is roughly 1.4, this answer is approx. 8.4. this is what we are looking for but we need to check to see if any other choices are larger than 8
B. too small
C. too small
D. too small
E. too small

Answer A!


This was majorly brilliant. I was going through a ridiculous quadratic equation only to think, "There has to be an easier way to solve this," and here it is.
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 13 Nov 2014, 20:51
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The ratio of the length to width of the yard = 5: 4

Let length = 5x and width = 4x

therefore we know , 5x * 4x = 15*12 - 5x*4x
=> x^2 = 4.5
Therefore, x = 3 * sqrt (1/2)

Therefore, width of the the yard = 4x = 4 * 3 * sqrt (1/2) = 6 * sqrt(2)

Hence A) is the answer
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 14 Nov 2014, 09:37
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Official Solution:

Image
Note: Figure not drawn to scale.

A rectangular yard (white area) is surrounded by a hedge (shaded region) with a length of 15 meters and a width of 12 meters, as shown in the figure above. The area of the yard is equal to the area of the hedge. If the ratio of the length of the hedge to the width of the hedge is the same as the ratio of the length of the yard to the width of the yard, what is the width, in meters, of the yard?

A. \(6\sqrt{2}\)
B. \(\frac{5}{4}\)
C. \(\frac{6}{\sqrt{2}}\)
D. \(7\frac{\sqrt{2}}{2}\)
E. \(\frac{6}{2}\)


Let the length of the yard be \(x\), and let the width of the yard be \(y\).

We are told that the length of the hedge is 15 meters and that the width of the hedge is 12 meters, so we can find the area of the yard and hedge combined using the formula for area of a rectangle: \(A_{\text{yard and hedge}} = \text{length} \times \text{width} = 15 \times 12 = 180\). Similarly, we can find the area of the yard: \(A_{yard} = xy\).

We are told that the area of the yard and the hedge are equal, so the area of the hedge must also be \(A_{hedge} = A_{yard} = xy\). We can use this information to write \(xy + xy = 2xy = 180\), or \(xy = 90\).

We know that the length and width of the yard are in the same ratio as the length and width of the hedge. We express this with a proportion: \(\frac{15}{12} = \frac{x}{y}\). Cross-multiply to get \(15y = 12x\). Simplify to get \(5y = 4x\).

Now we have two equations for \(x\) and \(y\). We will solve for these unknowns using substitution. Since we want to find the width, \(y\), we first solve for \(x\) in terms of \(y\). From the equation \(xy = 90\), we get \(x = \frac{90}{y}\). We plug this into \(5y = 4x\) and get \(5y = 4 \times \frac{90}{y} = \frac{360}{y}\). Dividing both sides by 5 and multiplying by \(y\), we find \(y^2 = 72\). We now take the square root of both sides to get \(y = \sqrt{72}\).

To simplify this square root, we look for a perfect square that is a factor of 72. We note that \(72 = 36 \times 2\), and so \(y = \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).

Answer: A.
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 14 Nov 2014, 21:46
I missed the fact that one should 180/2=90, so

5x*4x=90
20x^2=90
x^2=90/20 --> x=3*sqrt2/2

finally, 3*sqrt2*4/2=6*sqrt2

A
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 05 Jun 2015, 06:13
I don't get it. Why is everyone taking the length and breadth of the hedge as 15 and 12? Wouldn't the area be 180 then? And we know the area is 90. So, shouldn't we equate X/Y = 15-X/12-Y??
Can anyone tell me what silly mistake i am doing?
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 05 Jun 2015, 06:16
I don't get it. Why is everyone taking the length and breadth of the hedge as 15 and 12? Wouldn't the area be 180 then? And we know the area is 90. So, shouldn't we equate X/Y = 15-X/12-Y??
Can anyone tell me what silly mistake i am doing?
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 05 Jun 2015, 06:36
AnubhavRao wrote:
I don't get it. Why is everyone taking the length and breadth of the hedge as 15 and 12? Wouldn't the area be 180 then? And we know the area is 90. So, shouldn't we equate X/Y = 15-X/12-Y??
Can anyone tell me what silly mistake i am doing?
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The ratio of the length of the hedge (15) to the width of the hedge (12) is the same as the ratio of the length of the yard (x) to the width of the yard (y): 15/12 = x/y

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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 05 Jun 2015, 06:41
AnubhavRao wrote:
I don't get it. Why is everyone taking the length and breadth of the hedge as 15 and 12? Wouldn't the area be 180 then? And we know the area is 90. So, shouldn't we equate X/Y = 15-X/12-Y??
Can anyone tell me what silly mistake i am doing?
Bunuel
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 05 Jun 2015, 06:44
Ohhh so those aren't the actual dimensions but their ratios. Dimensions will be less than those so as to have an area of 90.
My bad.
Thank you for swift response, Bunuel :)
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 28 Dec 2015, 16:41
180-lw=lw or 180=2lw or lw=90.

now l/w=5/4

5l/4 * l = 90
l^2 = 90*4/5
l^2 = 72
l= sqrt(72)
sqrt(72) = 6(sqrt2)

A.
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 01 Jan 2016, 13:37
let L,W=length and width of hedge
let l,w=length and width of yard
LW=180m^2
lw=LW-lw=90m^2
15/12=5/4
(5x)(4x)=90
x^2=9/2
x=3/sqrt2
4x=(6)(sqrt2) meters width of yard
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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New post 11 May 2016, 10:20
A it is well explained above !
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Re: A rectangular yard (white area) is surrounded by a hedge (shaded regio  [#permalink]

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