SOLUTION: A brick patio is twice as long as it is wide. It is bordered on all sides by a garden 1.5m wide. Find the dimensions of the patio if the area of the garden is 54m^2.

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Question 174134: A brick patio is twice as long as it is wide. It is bordered on all sides by a garden 1.5m wide. Find the dimensions of the patio if the area of the garden is 54m^2.
Found 3 solutions by scott8148, checkley77, solver91311:
Answer by scott8148(6628) About Me  (Show Source):
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let x=width, so 2x=length

area = area __ (x+3)(2x+3) = (x)(2x) + 54

2x^2+9x+9 = 2x^2+54

subtracting 2x^2+9 __ 9x = 45

dividing by 9 __ x = 5

Answer by checkley77(12844) About Me  (Show Source):
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AREA OR THE BRICK PATIO=LW
L=2W
AREA=2W*W=2W^2
AREA OF GARDEN=54
54=(L+3)(W+3)-2W^2
54=(2W+3)(W+3)-2W^2
54=2W^2+9W+9-2W^2
54=9W+9
9W=54-9
9W=45
W=45/9
W=5m.
L=2*5=10m.
PROOF:
54=(10+3)(5+3)-2*5^2
54=13*8-2*25
54=104-50
54=54

Answer by solver91311(24713) About Me  (Show Source):
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Let the width of the patio be w, then the length of the patio is 2w. Since the garden is 1.5 meters wide, and the garden goes all the way around, the width of the rectangle formed by the outside edge of the garden must be w+%2B+1.5+%2B+1.5=w+%2B+3. Similarly, the length of the outside rectangle must be 2w+%2B+3.

The area of the brick patio is the length times the width or 2w%5E2 and the area of the entire rectangle including the garden and the patio is again length times width or %282w+%2B+3%29%28w+%2B+3%29+=+2w%5E2+%2B+9w+%2B+9.

We know the area of the garden part is 54 square meters and the area of the patio part is 2w%5E2, so the sum of these two quantities must be the overall area. Now that we have two expressions for the overall area, we can set them equal:

2w%5E2+%2B+9w+%2B+9+=+2w%5E2+%2B+54

Add -2w%5E2 to both sides:

9w+%2B+9+=+54

Add -9 to both sides:

9w=45

Finally, multiply both sides by 1%2F9

w+=+5 meaning the width of the patio is 5 meters.

Since the length of the patio is 2w, the length must be 10 meters.

Check:

Overall area: 2w%5E2+%2B+9w+%2B+9=2%285%29%5E2%2B9%285%29%2B9=50%2B45%2B9=104

Area of the patio: 2w%5E2=2%285%29%5E2=50

Overall area minus the area of the patio must be equal to the area of the garden. 104-50=54, and 54 square meters was the given area of the garden. Answer checks.