SOLUTION: Mr. Garcia owns an irregular field that is in the shape of two squares, side by side. The area of the field is 17,396 yd2, and the perimeter is 572 yd. What is the length of the lo

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Question 1194689: Mr. Garcia owns an irregular field that is in the shape of two squares, side by side. The area of the field is 17,396 yd2, and the perimeter is 572 yd. What is the length of the longest side of the field?
Found 3 solutions by math_tutor2020, greenestamps, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

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: 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








which is approximate

or

or

or
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.
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


An "irregular field that is in the shape of two squares, side by side" suggests that the side lengths of the two squares are different; so the field looks something like this:

  +-----------+
  |           |
  |           +-----+
  |           |     |
  |           |     |
  +-----------+-----+


or perhaps this:
  +-----------+
  |           +-----+
  |           |     |
  |           |     |
  |           +-----+
  +-----------+


Let x and y be the side lengths of the larger and smaller square, respectively.

Then the combined area of the two squares is



The perimeter of the field is



So we need to solve a pair of simultaneous equations:

(1)
(2)

You can solve the pair of equations by solving (2) for y and substituting in (1); you will get an ugly quadratic equation whose solutions you would probably need a calculator for.

Since a calculator was going to be necessary, I solved the problem by a different path.

I used the TABLE feature of a TI-83 calculator to find integer solution(s) of

-->

and found there was only one: x=100 and y=86. And that satisfied the condition that the perimeter 4x+2y is 572.

From there, I can't answer the question that was asked. If the field looks like the first diagram above, the longest side of the field is 100+86=186 yards; if it looks like the second diagram, the longest side of the field is 100 yards.

ANSWER: The longest side of the field is either 100 yards or 186 yards
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
Mr. Garcia owns an irregular field that is in the shape of two squares, side by side. The area of the field is 17,396 yd2, and the perimeter is 572 yd. What is the length of the longest side of the field?
 .
Let each side of the smaller and larger squares, be S and L, respectively
Then areas of the smaller and larger squares = S2 and L2, respectively
Also, perimeter of smaller square = 2S (side-by-side), while perimeter of larger square = 4L

With area of field (both squares) being 17,396 sq yds, we get: 
Also, with perimeter of field (2 squares side-by-side) being 572 yds, we get 4L + 2S = 572 
2(2L + S) = 2(286)______2L + S = 286____S = 286 - 2L ------- eq (ii)

                 ------ Substituting 286 - 2L for S in eq (i)

    5L(L - 100) - 644(L - 100) = 0
           (5L - 644)(L - 100) = 0
Each side of larger square, or 

If each side of larger square = , S, or each smaller square’s side = 

If each side of larger square = 100 yds, S, or each smaller square’s side = 286 - 2(100) = 286 - 200 = 86 yds

Therefore, with the given info., the longer side of the field can either be: 

To get a more definitive answer, certain restrictions would need to be placed on at least one of the squares. For example, if the larger square needs
to occupy the entire width of the field, then the longer  of the 2 possible sides would be used for the larger square (see Diagram A).

However, if for example, a path needs to be placed around the larger square, then it'd be best to go with the shorter of the 2 larger-square's sides, 
which is 100 yds (see Diagram B). 
It could also be stated that the smaller square needs to have the largest possible area. In this case, you'd go with the longer of the 2 smaller-square's
sides, or 86 yds (see Diagram B), as opposed to .

Note that the PLACEMENT of the smaller square on ANY side of the larger square DOES NOT matter! This means that it could be in the center, towards the top,
at the very bottom, or anywhere else on one of the larger square's sides!

You can do the CHECK!!
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