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The figure shows a square patio surrounded by a walkway of

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The figure shows a square patio surrounded by a walkway of [#permalink] New post 06 Feb 2006, 21:26
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?
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 [#permalink] New post 06 Feb 2006, 22:58
64.

I'll post the solution if the answer is correct.
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 [#permalink] New post 06 Feb 2006, 23:12
Solution:

Divide the Area of the Pathway into 4 sections. 2 of them will have the area of (x)(5+x) and 2 of them will have (x)(5+3x).

Area of Pathway = 2(x)(5+x) + 2(x)(5+3x) = 132

Since it is a Square Patio, all sides = 5+x.

Area of Square Patio = (5+x)(5+x)

Solve for x.

10x + 2x^2 + 10x + 6x^2 = 132

20x + 8x^2 = 132

8x^2 + 20x - 132 = 0

2x^2 + 5x - 33 = 0

Use Quadratic Formula

x = {-5 +/- sqrt[5^2 - 4(2)(-33)]}/2(2)

x = (-5 +/- 17)/4

x = -5.5 or 3 (x is therefore 3 because you cannot have negative #s)

Plug-back

Area of Square Patio = (5+x)(5+x) = 8(8) = 64
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 [#permalink] New post 06 Feb 2006, 23:19
thanks coolfish. but is there a way to keep it simple and not use the quadratic formula?

here is how i started

bigger square: w=8x area=64x^2

smaller square= (5+x)^2

64x^2-(5+x^2)=132

then i just blanked. (poor math skills)
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 [#permalink] New post 06 Feb 2006, 23:36
I think you made a mistake, the width of the bigger square is (5+3x), not 8x.

It is possible to not use the Quadratic Formula, just takes some time to do number plug-ins. I did Quadratic Formula cause I do not want to deal with the plug-ins since there is a 2x.

Another way of solving this problem would be:

Big Square - Pathway = Small Square

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

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

8x^2+20x-132 = 0

2x^2+5x-33 = 0

Try numbers for plug-in so the sum will added up to 5x, this one is actually not very difficult, after I tried it.

(2x+11)(x-3) = 0 [11x - 6x = 5x]

x = -5.5 or 3

(x+5)(x+5) = 8(8) = 64
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 [#permalink] New post 06 Feb 2006, 23:46
64 sq meter

if you extrapolate the corners of the patio to the walkaway line you can determine the area of the whole walk away.

such that (x+5)*x*4 + x*x *4 =132

by solving x=3,

so the side of the square is 5+3 = 8
area 8*8= 64

though it took more than 2 min to solve!
  [#permalink] 06 Feb 2006, 23:46
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