SOLUTION: Don't know how to set up this one. 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 th

Algebra ->  Coordinate-system -> SOLUTION: Don't know how to set up this one. 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 th      Log On


   

Question 149797: Don't know how to set up this one.

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?

Thank you for your help. Your assistance is greatly appreciated.
Answer by nerdybill(7384) About Me  (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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"area bordered by the outside edge of the path" minus "area of garden" equals "area of the path"
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Let x = width of path
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"area bordered by the outside edge of the path" = (18+2x)(13+2x)
"area of garden" = 18*13
"area of the path" = 516
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(18+2x)(13+2x) - (18*13) = 516
234 + 26x + 36x + 4x^2 - 234 = 516
62x + 4x^2 = 516
4x^2 + 62x - 516 = 0
x^2 + 13x - 129 = 0
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Can't be factored easily so you must use the quadratic equation. Doing so, will produce a positive and a negative solution. Since a negative answer doesn't make sense, the positive solution will be your answer.
x = 6.586 feet
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Below is the quadratic solution:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B13x%2B-129+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2813%29%5E2-4%2A1%2A-129=685.

Discriminant d=685 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-13%2B-sqrt%28+685+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2813%29%2Bsqrt%28+685+%29%29%2F2%5C1+=+6.5862523283024
x%5B2%5D+=+%28-%2813%29-sqrt%28+685+%29%29%2F2%5C1+=+-19.5862523283024

Quadratic expression 1x%5E2%2B13x%2B-129 can be factored:
1x%5E2%2B13x%2B-129+=+1%28x-6.5862523283024%29%2A%28x--19.5862523283024%29
Again, the answer is: 6.5862523283024, -19.5862523283024. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B13%2Ax%2B-129+%29