Question 1194689
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Edit: Disregard my answer below. The tutor greenestamps makes a good point that the squares are very likely of different sizes, which I didn't consider. My apologies.


It sounds like this field is a rectangle, since the two squares are joined side by side (I'm assuming on the same horizontal level).
I don't know why your teacher considers a rectangle to be irregular, unless s/he means "the figure isn't a regular polygon".


Regular polygon = it has all sides equal, and all angles equal
Examples: <ul><li>an equilateral triangle is a regular polygon with 3 equal sides. All angles are 60 degrees.</li><li>any square is a four sided regular polygon. All four sides are equal, and all four angles are equal to 90 degrees.</li></ul>Anyways, onto the problem at hand.


L = length = longer side
W = width = shorter side
These are positive real numbers.


P = perimeter of a rectangle
P = 2(L+W)


Plug in the given perimeter P = 572 and solve for L
P = 2(L+W)
572 = 2(L+W)
L+W = 572/2
L+W = 286
L = 286-W


A = area of the rectangle
A = length*width
A = L*W
A = (286-W)*W


Plug in the given area A = 17396 and rearrange terms like so
The goal from here is to solve for W.
A = (286-W)*W
17396 = (286-W)*W
17396 = 286W-W^2
W^2-286W+17396 = 0


This may be able to be factored, but the trial and error process is quite frankly annoying. It may turn out that this cannot be factored at all.


I prefer the quadratic formula instead.
It is more direct and to the point. 
No guess-and-check is required, and it works for any quadratic.


Plug in a = 1, b = -286, c = 17396
{{{W = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{W = (-(-286)+-sqrt((-286)^2-4(1)(17396)))/(2(1))}}}


{{{W = (286+-sqrt(81796 - 69584))/(2)}}}


{{{W = (286+-sqrt(12212))/(2)}}}


{{{W = (286+-  110.507918)/(2)}}} which is approximate


{{{W = (286+110.507918)/(2)}}} or {{{W = (286-110.507918)/(2)}}}


{{{W = (396.507918)/(2)}}} or  {{{W = (175.492082)/(2)}}}


{{{W = 198.253959}}} or  {{{W = 87.746041}}}
The decimal values are approximate.


If W = 198.253959, then,
L = 286-W = 286-198.253959 = 87.746041


Then notice how
area = L*W = 87.746041*198.253959 = 17,396.0000148263
which is fairly close to the target area of 17,396.
This helps confirm the answer.
Use more precision in the quadratic formula steps to get a more accurate value for W, which in turn leads to a more accurate value for L.


You should find that if W = 87.746041, then it leads to L = 198.253959


Answer: The longest side is roughly equal to 198.253959 yards.
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