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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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\n" ); document.write( "Draw a diagram of the problem -- it'll help you see the problem.
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\n" ); document.write( "Area of lawn:
\n" ); document.write( "(50-2x)(120-2x)
\n" ); document.write( "= 2(25-x)2(60-x)
\n" ); document.write( "= 4(25-x)(60-x)
\n" ); document.write( "= 4(150-85x+x^2)
\n" ); document.write( "= 4x^2 - 340x + 6000
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\n" ); document.write( "Area of path:
\n" ); document.write( "(50)(120) - (50-2x)(120-2x)
\n" ); document.write( "= 6000 - (4x^2 - 340x + 6000)
\n" ); document.write( "= -4x^2 + 340x
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\n" ); document.write( "Since:
\n" ); document.write( "\"Area of lawn\" = \"Area of path\"
\n" ); document.write( "we have
\n" ); document.write( "4x^2 - 340x + 6000 = -4x^2 + 340x
\n" ); document.write( "8x^2 - 340x + 6000 = 340x
\n" ); document.write( "8x^2 - 680x + 6000 = 0
\n" ); document.write( "x^2 - 85x + 750 = 0
\n" ); document.write( "(x-75)(x-10) = 0
\n" ); document.write( "x = {10, 75}
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\n" ); document.write( "We can toss out the 75 leaving:
\n" ); document.write( "x = 10 meters (width of path)
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\n" ); document.write( "dimensions of lawn:
\n" ); document.write( "width = 50 -2x = 50 - 20 = 30 meters
\n" ); document.write( "length = 120 -2x = 120 - 20 = 100 meters
\n" ); document.write( "Therefore, the lawn is
\n" ); document.write( "100 x 20 meters \n" ); document.write( "