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A square garden is surrounded by a path of uniform width. If the path

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Bunuel
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A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   29 Aug 2017, 02:07
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A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x?


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

B. \(2 \sqrt{x} - \sqrt{2}\)

C. \(\frac{\sqrt{2}}{2} - \frac{x}{4}\)

D. \(x \sqrt{2} - \frac{x}{2}\)

E. \(\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}\)
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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   01 Sep 2017, 12:09
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Bunuel wrote:
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x?


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

B. \(2 \sqrt{x} - \sqrt{2}\)

C. \(\frac{\sqrt{2}}{2} - \frac{x}{4}\)

D. \(x \sqrt{2} - \frac{x}{2}\)

E. \(\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}\)
\

Since the square garden has an area of x, its side length is √x. Since the square garden is surrounded by a path of uniform width, the shape of the path and garden combined is also a square. We can let the width of the path = n, and thus the side length of the square that is the path and garden combined is √x + 2n. Since the total area of the path and garden is x + x = 2x, we have:

(√x + 2n)^2 = 2x

Taking the square root of both sides, we have:

√x + 2n = √(2x)

2n = √(2x) - √x

n = √(2x)/2 - √x/2

Answer: E
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A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   29 Aug 2017, 02:38
Bunuel wrote:
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x?


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

B. \(2 \sqrt{x} - \sqrt{2}\)

C. \(\frac{\sqrt{2}}{2} - \frac{x}{4}\)

D. \(x \sqrt{2} - \frac{x}{2}\)

E. \(\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}\)


consider x as 4 then side of square will be 2
now side of square +path = \(2\sqrt{2}\)
width = \((2\sqrt{2} - 2)/2\) = \(\sqrt{2}-1\)
substituting 4 in place of x in options
E
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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   30 Aug 2017, 11:01
Luckisnoexcuse wrote:
Bunuel wrote:
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x?


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

B. \(2 \sqrt{x} - \sqrt{2}\)

C. \(\frac{\sqrt{2}}{2} - \frac{x}{4}\)

D. \(x \sqrt{2} - \frac{x}{2}\)

E. \(\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}\)


consider x as 4 then side of square will be 2
now side of square +path = \(2\sqrt{2}\)
width = \((2\sqrt{2} - 2)/2\) = \(\sqrt{2}-1\)
substituting 4 in place of x in options
E


How now side of square +path = \(2\sqrt{2}\)???

Thanks
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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   01 Sep 2017, 10:01
Bunuel wrote:
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x?


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

B. \(2 \sqrt{x} - \sqrt{2}\)

C. \(\frac{\sqrt{2}}{2} - \frac{x}{4}\)

D. \(x \sqrt{2} - \frac{x}{2}\)

E. \(\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}\)


E is the answer
the area of the garden is 2x
side of the garden is square root of 2x
the shaded area is a trapezoid, its area equal 1/4 the area of the path, and its height is the width of the path.
you can see in the picture attached
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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink] New post   06 Oct 2022, 10:30
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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink]
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