A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of \(x,\) then

[Vincen]

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. \(\sqrt2x\)

B. \(2\sqrt{x}-\sqrt2\)

C. \(\dfrac{\sqrt2}2-\dfrac{x}4\)

D. \(\sqrt2x-\dfrac{x}2\)

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

Answer: E

Source: Magoosh

[Scott@TargetTestPrep]

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. \(\sqrt2x\)

B. \(2\sqrt{x}-\sqrt2\)

C. \(\dfrac{\sqrt2}2-\dfrac{x}4\)

D. \(\sqrt2x-\dfrac{x}2\)

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

Answer: E

Solution:

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