SOLUTION: A flower bed is in the shape of a rectangle. Its length is twice its width. The bed is surrounded by a walkway 4 feet wide. If the area of the walkway is exactly twice the area

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Question 4462: A flower bed is in the shape of a rectangle. Its length is twice its width. The bed is surrounded by a walkway 4 feet wide. If the area of the walkway is exactly twice the area of the bed, find the dimensions of the bed.
Answer by rapaljer(4671) About Me  (Show Source):
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Begin by drawing a rectangle whose width is x, and whose length is 2x. Add another rectangle outside this rectangle that extends 4 feet in each direction from the inner rectangle. The width of the outer rectangle is x+8, and the length of the outer rectangle is 2x+8, since you have to add 4 feet TWICE in each direction.

The area of the inner rectangle is LW = x(2x) or 2x%5E2.
The area of the outer rectangle is LW = (x+8)(2x+8) or 2x%5E2+%2B+24x+%2B+64.
Area of walkway = outer area - inner area
= 2x%5E2+%2B+24x+%2B+64+-+2x%5E2
= 24x+%2B+64

Equation: Area of walkway equals twice inner area (area of the bed):
24x+%2B+64=+2%2A%282x%5E2%29 .
24x+%2B+64+=+4x%5E2

Quadratic equations, so set equal to zero by subtracting 24x and 64 from both sides.
0+=+4x%5E2+-+24x+-+64+=+0

Factor out the common factor of 4:
0=4%28x%5E2+-+6x+-+16%29+

Factor the trinomial:
0+=+4%28x-8%29%28x%2B2%29+

Solutions are x = 8 and x = -2, but x represents the side of a rectangle, so it cannot be negative.

Final answer:
x = 8 width of bed.
2x = 16 length of bed.

Check:
Dimensions of the walkway
x+8 = 16
2x+8= 24
Area of bed = 8%2A16 = 128 sq. ft.
Total area = 16%2A24 = 384 sq. ft.
Area of walk = 384 - 128 = 256 sq. ft.

Area of walk = 2(area of bed)
256 = 2(128)

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