SOLUTION: A square garden has a walkway 3 ft wide around its outer edge.If the area of the entire garden, including the walkway, is 18,000 ft^2, what are the dimensions of the planted area?

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Question 1189792: A square garden has a walkway 3 ft wide around its outer edge.If the area of the entire garden, including the walkway, is 18,000 ft^2, what are the
dimensions of the planted area?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

x = side length of the planted area (ie we ignore the walkway)

When considering the walkway, we add on 3+3 = 6 feet for each dimension of the square.
The x by x square bumps up to (x+6) by (x+6)

The larger square has area of
(x+6)^2
(x+6)*(x+6)
x(x+6)+6(x+6)
x^2+6x+6x+36
x^2+12x+36

Set this area expression equal to the desired area (18,000) and solve for x.
x^2+12x+36 = 18,000
x^2+12x+36-18,000 = 0
x^2+12x-17,964 = 0

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

x+=+%28-%2812%29%2B-sqrt%28%2812%29%5E2-4%281%29%28-17964%29%29%29%2F%282%281%29%29

x+=+%28-12%2B-sqrt%2872000%29%29%2F%282%29

x+=+%28-12%2B-268.3281573%29%2F%282%29

x+=+%28-12%2B268.3281573%29%2F%282%29 or x+=+%28-12-268.3281573%29%2F%282%29

x+=+%28256.3281573%29%2F%282%29 or x+=+%28-280.32815728%29%2F%282%29

x+=+128.16407865 or x+=+-140.16407863

We ignore the negative x value because we cannot have negative side lengths.

The square planted area is roughly 128.16 feet by 128.16 feet.

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Here's another way to solve.

We start off with some smaller square representing the plants only. Adding on the walkway of uniform with around all the edges will get us some larger square.

Let's say this larger square has side lengths of W. So W^2 is the area of this larger square.
Set it equal to the 18,000 figure and solve for W

W%5E2+=+18000

W+=+sqrt%2818000%29

W+=+134.164079

The side lengths of the larger square are roughly 134.164079 feet

We then subtract off 6 feet due to the 3+3 = 6 figure calculated earlier
134.164079 - 6 = 128.164079

We get the same result of 128.16 as earlier.

Depending on your preferences, this option may be much easier compared to the last method.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
A square garden has a walkway 3 ft wide around its outer edge.
If the area of the entire garden, including the walkway, is 18,000 ft^2, what are the
dimensions of the planted area?
~~~~~~~~~~~~~~~ 

Notice that under given conditions, the exterior shape is a square of the area 18000 ft^2.


Hence, the side of the exterior square is  sqrt%2818000%29 = 134.16 feet.


It means that the planted square side length is  134.16 - 3 - 3 = 128.16 feet.    ANSWER

Solved.