SOLUTION: A 50 m by 120 m park consists of a rectangular lawn surrounded by a path of uniform width. Find the dimensions of the lawn if its area is the same as the area of the path. (Hint: L

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Question 198087: A 50 m by 120 m park consists of a rectangular lawn surrounded by a path of uniform width. Find the dimensions of the lawn if its area is the same as the area of the path. (Hint: Let x = the width of path)
Answer by nerdybill(7384) About Me  (Show Source):
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A 50 m by 120 m park consists of a rectangular lawn surrounded by a path of uniform width. Find the dimensions of the lawn if its area is the same as the area of the path. (Hint: Let x = the width of path)
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Draw a diagram of the problem -- it'll help you see the problem.
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Area of lawn:
(50-2x)(120-2x)
= 2(25-x)2(60-x)
= 4(25-x)(60-x)
= 4(150-85x+x^2)
= 4x^2 - 340x + 6000
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Area of path:
(50)(120) - (50-2x)(120-2x)
= 6000 - (4x^2 - 340x + 6000)
= -4x^2 + 340x
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Since:
"Area of lawn" = "Area of path"
we have
4x^2 - 340x + 6000 = -4x^2 + 340x
8x^2 - 340x + 6000 = 340x
8x^2 - 680x + 6000 = 0
x^2 - 85x + 750 = 0
(x-75)(x-10) = 0
x = {10, 75}
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We can toss out the 75 leaving:
x = 10 meters (width of path)
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dimensions of lawn:
width = 50 -2x = 50 - 20 = 30 meters
length = 120 -2x = 120 - 20 = 100 meters
Therefore, the lawn is
100 x 20 meters