SOLUTION: A 10m by 34m garden is surrounded by a walkway of uniform width (call it "x"). The total area of the garden and walkway is 640 square meters. what is the width of the walkway?

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Question 165093: A 10m by 34m garden is surrounded by a walkway of uniform width (call it "x"). The total area of the garden and walkway is 640 square meters. what is the width of the walkway?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = width of the path


With these type of problems, it helps to draw a picture. So draw two rectangles, one inside of the other, and label the inside rectangle's dimensions:




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Now the label the width of the path "x" (denoted in red)

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Since there are 2 "x" lengths per side, this means that you need to add "2x" to both the length and width of the inner rectangle to get the length and width of the outer rectangle. If this makes no sense at all, here's a visual:


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So the length and width of the outer rectangle is 34%2B2x and 10%2B2x respectively. This means that for the outer rectangle A=640 (the given area of both the walkway and the garden), L=34%2B2x and W=10%2B2x


Remember, the area of any rectangle is A=LW


A=LW Start with the area of a rectangle formula


640=%2834%2B2x%29%2810%2B2x%29 Plug in A=640, L=34%2B2x and W=10%2B2x


640=340%2B68x%2B20x%2B4x%5E2 FOIL


0=340%2B68x%2B20x%2B4x%5E2-640 Subtract 640 from both sides.


0=4x%5E2%2B88x-300 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=4, b=88, and c=-300


Let's use the quadratic formula to solve for x.


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%2888%29+%2B-+sqrt%28+%2888%29%5E2-4%284%29%28-300%29+%29%29%2F%282%284%29%29 Plug in a=4, b=88, and c=-300


x+=+%28-88+%2B-+sqrt%28+7744-4%284%29%28-300%29+%29%29%2F%282%284%29%29 Square 88 to get 7744.


x+=+%28-88+%2B-+sqrt%28+7744--4800+%29%29%2F%282%284%29%29 Multiply 4%284%29%28-300%29 to get -4800


x+=+%28-88+%2B-+sqrt%28+7744%2B4800+%29%29%2F%282%284%29%29 Rewrite sqrt%287744--4800%29 as sqrt%287744%2B4800%29


x+=+%28-88+%2B-+sqrt%28+12544+%29%29%2F%282%284%29%29 Add 7744 to 4800 to get 12544


x+=+%28-88+%2B-+sqrt%28+12544+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%28-88+%2B-+112%29%2F%288%29 Take the square root of 12544 to get 112.


x+=+%28-88+%2B+112%29%2F%288%29 or x+=+%28-88+-+112%29%2F%288%29 Break up the expression.


x+=+%2824%29%2F%288%29 or x+=++%28-200%29%2F%288%29 Combine like terms.


x+=+3 or x+=+-25 Simplify.


So the possible answers are x+=+3 or x+=+-25


However, since a negative width is not possible, this means that x+=+-25 is NOT a solution.


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Answer:

So the solution is x+=+3 which means that the width of the path is 3 meters.