SOLUTION: I've racked my brain for hours & I still don't know where to begin on this problem: A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions

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Question 90931This question is from textbook Beginning Al
: I've racked my brain for hours & I still don't know where to begin on this problem: A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions are 30 ft by 40 ft. the total area, garden plus walkway, is to be 1800 ft^2. What must be the width of the walkway to the nearest thousandth?
Thank you. This question is from textbook Beginning Al

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If we draw a picture we get



If we let x be the width of the path, notice there are 2 x-values per side. So we add 2x to 40 to get 40+2x. This is the total length of the side that contains 40 feet.

Also, this means we add 2x to 30 to get 30+2x. This is the total length of the side that contains 30 feet.


Start with the general area function:
A=xy

1800=xy Plug in A=1800 (this is the total area)


1800=%2840%2B2x%29%2830%2B2x%29 Plug in x=40%2B2x and y=30%2B2x


1800=4x%5E2%2B140x%2B1200 Foil


0=4x%5E2%2B140x-600 Subtract 1800 from both sides



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


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 4%2Ax%5E2%2B140%2Ax-600=0 ( notice a=4, b=140, and c=-600)

x+=+%28-140+%2B-+sqrt%28+%28140%29%5E2-4%2A4%2A-600+%29%29%2F%282%2A4%29 Plug in a=4, b=140, and c=-600



x+=+%28-140+%2B-+sqrt%28+19600-4%2A4%2A-600+%29%29%2F%282%2A4%29 Square 140 to get 19600



x+=+%28-140+%2B-+sqrt%28+19600%2B9600+%29%29%2F%282%2A4%29 Multiply -4%2A-600%2A4 to get 9600



x+=+%28-140+%2B-+sqrt%28+29200+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F8 Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

x+=+%28-140+%2B+20%2Asqrt%2873%29%29%2F8 or x+=+%28-140+-+20%2Asqrt%2873%29%29%2F8


Now break up the fraction


x=-140%2F8%2B20%2Asqrt%2873%29%2F8 or x=-140%2F8-20%2Asqrt%2873%29%2F8


Simplify


x=-35+%2F+2%2B5%2Asqrt%2873%29%2F2 or x=-35+%2F+2-5%2Asqrt%2873%29%2F2


So these expressions approximate to

x=3.86000936329383 or x=-38.8600093632938


So our possible solutions are:
x=3.86000936329383 or x=-38.8600093632938

Since a negative length doesn't make sense, our only solution is x=3.86000936329383

which is 3.860 to the nearest thousandth


Check:

1800=%2840%2B2x%29%2830%2B2x%29 Start with the given area function

1800=%2840%2B2%283.860%29%29%2830%2B2%283.860%29%29 plug in x=3.860

1800=%2840%2B7.72%29%2830%2B7.72%29 multiply

1800=%2847.72%29%2837.72%29 Add

1800=1799.9984 Since we rounded, this is as close as it gets. So our answer is verified.