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A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet?
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\n" ); document.write( "Draw a rectangle representing the 18 by 13 ft garden, then draw a larger rectangle
\n" ); document.write( "around that one, enclosing the path around the garden.
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\n" ); document.write( "Label the width of the path as x
\n" ); document.write( "It will be apparent that the overall dimensions will be (18+2x) by (13+2x)
\n" ); document.write( "FOIL this to get the overall area:
\n" ); document.write( "(18+2x)*(13+2x) = 234 + 36x + 26x + 4x^2 = 234 + 62x + 4x^2
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\n" ); document.write( "Garden area; 18 * 13 = 234
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\n" ); document.write( "The equation:
\n" ); document.write( "Overall area - garden area = path area (given as 516 sq/ft)
\n" ); document.write( "(4x^2 + 62x + 234) - 234 = 516
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\n" ); document.write( "4x^2 + 62x - 516 = 0; our old friend, the quadratic equation
\n" ); document.write( "Simply divide equation by 2:
\n" ); document.write( "2x^2 + 31x - 258 = 0
\n" ); document.write( "Factor this to:
\n" ); document.write( "(2x + 43)(x - 6) = 0
\n" ); document.write( "Positive solution
\n" ); document.write( "x = +6 ft is the width of the path
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\n" ); document.write( "We can check this: overall dimensions will be (18+12) by (13+12)
\n" ); document.write( "(30*25) - 234 =
\n" ); document.write( "750 - 234 = 516, confirms our solution
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\n" ); document.write( "Did this make sense? Any questions?
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