Question 149541: 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?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! 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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Draw a rectangle representing the 18 by 13 ft garden, then draw a larger rectangle
around that one, enclosing the path around the garden.
:
Label the width of the path as x
It will be apparent that the overall dimensions will be (18+2x) by (13+2x)
FOIL this to get the overall area:
(18+2x)*(13+2x) = 234 + 36x + 26x + 4x^2 = 234 + 62x + 4x^2
:
Garden area; 18 * 13 = 234
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The equation:
Overall area - garden area = path area (given as 516 sq/ft)
(4x^2 + 62x + 234) - 234 = 516
:
4x^2 + 62x - 516 = 0; our old friend, the quadratic equation
Simply divide equation by 2:
2x^2 + 31x - 258 = 0
Factor this to:
(2x + 43)(x - 6) = 0
Positive solution
x = +6 ft is the width of the path
:
:
We can check this: overall dimensions will be (18+12) by (13+12)
(30*25) - 234 =
750 - 234 = 516, confirms our solution
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Did this make sense? Any questions?
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