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

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A square garden is surrounded by a path of uniform width. If the path [#permalink]

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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. $$\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}$$
[Reveal] Spoiler: OA

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A square garden is surrounded by a path of uniform width. If the path [#permalink]

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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]

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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 = $$2\sqrt{2}$$???

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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink]

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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. $$\frac{\sqrt{2x}}{2} - \frac{\sqrt{x}}{2}$$

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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FullSizeRender 7.jpg [ 1.09 MiB | Viewed 338 times ]

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Re: A square garden is surrounded by a path of uniform width. If the path [#permalink]

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01 Sep 2017, 12:09
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

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Re: A square garden is surrounded by a path of uniform width. If the path   [#permalink] 01 Sep 2017, 12:09
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