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The figure shows a square patio surrounded by a walkway of width x

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sanket1991

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20 Jul 2015, 04:15

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width of the walkway = x
width of the small square = x+5 = x+5
width of the whole large square = x+5+x+x = 3x+5

Difference between area
(3x+5)^2 - (x+5)^2 = 132
8x^2 +20x = 132
4x(2x+5) = 132
x (2x+5) = 33 = 3*11
Hence, x = 3
x+5 = 8
8^2 = 64

anik19890

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Engr2012

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amitdgr

The figure shows a square patio surrounded by a walkway of width x meters. If the area of the walkway is 132 square meters and the width of the patio is 5 meters greater than the width of the walkway, what is the area of the patio, in square meters?

A. 56
B. 64
C. 68
D. 81
E. 100

Hi bunuel,

Could you please provide your comments on how we are getting 5+3x.

i didnt get it.

Thanks.

It is given that width of the patio is 5 more than width of the walkway. Now, walkway's 'extension on either side of the patio' is 'x' and thus width of the of the patio = $W_P$ = 5+x

Now, total width of the walkway = $W_W$ = width of the patio + 2*x = 5+x+2x = 5+3x

it requires a redesign figure, which is not provided by anyone, only for this reason many confusion arise, could you please make a understandable figure please?

Manonamission

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22 Oct 2016, 05:20

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(3x+5)^2 - (x+5)^2 = 132

Solving for x = 64

Hence B

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19 Jul 2017, 00:11

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I eventually didn't understand the figure . The outer box is the walkway and the inner one is the patio. Am i correct? Sorry seems like silly question, but iam weak in geometry and it would really great if someone could help me understand the figure.

Bunuel

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19 Jul 2017, 01:03

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longhaul123

We are given that the patio is surrounded by a walkway, so yellow area is walkway and red is patio itself.

Attachment:

patio.JPG [ 13.39 KiB | Viewed 2705 times ]

longhaul123

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19 Jul 2017, 23:01

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Thank youBunuel

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I did not realise this when I first read the question but with hindsight the fastest way to solve this is to realise that:

both the answer and answer+132 must be perfect squares so:

A) 56 not a square
B) 64 is ok and 64+132 =196 is also ok
C) 58 not a square
D) 81+132 = 213 not a square
E) 100+132 = 232 not a square

so answer is C

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25 Jul 2017, 11:14

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rthothad

If we let x = the width of the walkway, then the width of the patio = x + 5. Let’s create the equation for the area of the combined walkway and patio. We see that the entire length of the combined walkway-patio is (x + x + x + 5) = (3x + 5). Because it is a square, we square (3x + 5) to determine the combined area. We see that this is equal to the area of the patio plus the area of the walkway:

(3x + 5)^2 = (x + 5)^2 + 132

9x^3 + 30x + 25 = x^2 + 10x + 25 + 132

8x^2 + 20x - 132 = 0

2x^2 + 5x - 33 = 0

(2x + 11)(x - 3) = 0

2x + 11 = 0

2x = -11

x = -5.5

Or x = 3

Since x cannot be negative, x = 3.

Since x = 3, the area of the patio is (3 + 5)^2 = 8^2 = 64.

Answer: B

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07 Aug 2019, 22:39

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rthothad

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Attachment:

patio.JPG

Leave everything else.
If I have to add something to 132 the result should be a perfect square. Option B works.

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07 Aug 2019, 23:50

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Since it is a square the area should be a perfect square.

Area given 132

Option b 64

Total area 64+132=196

Side =14

Only this option has a perfect square

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08 Aug 2019, 00:09

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Ans 64. Attached solution

Posted from my mobile device

Attachments

IMG_20190808_123756099.jpg [ 2.43 MiB | Viewed 1983 times ]

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10 Jul 2020, 12:13

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rthothad

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Attachment:

patio.JPG

132+64 = 196
ONLY AVAILABLE CHOICE WHEN ADDED TO 132 GIVES PERFECT SQUARE
ANS:B

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rthothad

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Attachment:

patio.JPG

We can PLUG IN THE ANSWERS, which represent the area of the patio.
Since the area of the walkway = 132, the correct answer must yield the following:
outer square - patio = 132

D: 81
Since the area of the patio = 81, each side of the patio = 9 meters
Since the width of the walkway is 5 less than the width of the patio, the width of the walkway = 9-5 = 4 meters
Since the walkway extends 4 meters to the left of the patio and another 4 meters to the right of the patio, the width of the outer square = 4+9+4 = 17, implying that the area of the outer square $= 17^2 = 289$
Resulting difference:
outer square - patio = 289-81 = 208
The difference is too great.
Eliminate D.

B: 64
Since the area of the patio = 64, each side of the patio = 8 meters
Since the width of the walkway is 5 less than the width of the patio, the width of the walkway = 8-5 = 3 meters
Since the walkway extends 3 meters to the left of the patio and another 3 meters to the right of the patio, the width of the outer square = 3+8+3 = 14, implying that the area of the outer square $= 14^2 = 196$
Resulting difference:
outer square - patio =196-64 = 132
Success!

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baraa900

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[quote="LiveStronger"]B

(5+3x)^2 - (5+x)^2 = 132
2x(10+4x) = 132
x(5+2x) = 33 = 3*11
So, x can be 3 and not 11

8^2 = 64[/q

how is the side of the large cube is 5?

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Area of Walkway = (Area of Entire Square) - (Area of Inner Patio Square)

Side Length of Inner Square Patio ------> the width/side is +5 more than the width X of the patio -----> A = (X + 5)

Side Length of ENTIRE Square ------> (X + 5) + X + X = (3X + 5)

Area of Walkway = (3X + 5)^2 - (X + 5)^2 = 132

(9X^2 + 30X + 25) - (X^2 + 10X + 25) = 132

8X^2 + 20X = 132

2X^2 + 5X = 33

2(x)^2 + 5(x) - 33 = 0

2(x)^2 - 6x + 11x - 33 = 0

2x(x-3) + 11(x -3) = 0

(x - 3) * (2x + 11) = 0

Since the Width = X can NOT be (-)Negative, the Width X must = +3

Area of Patio is given by ----> (X + 5)^2 = (3 + 5)^2 =

64

-B-

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AJB77

Clearly the patio is a square, since all of its sides of off 2x from the bigger square.

Let the patio's side be a. => x=a-5

Side of bigger square = a + 2(a-5) = 3a - 10

The area of the walkway = Area of big square - Area of patio

132 = (3a-10)^2 - a^2

132 = 9a^2 -60a + 100 -a^2

This reduces to:

8a^2 - 60a - 32 = 0
2a^2 -15a - 8 = 0
2a^2 + a -16a - 8 = 0
a(2a + 1) -8(2a+1) = 0
(a-8)*(2a+1) = 0
a = 8 or -1/2, But a is positive => a=8

Patio's area = a^2 = 64

B is my answer.

Thanks for the explaination

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