SOLUTION: A soccer field has a perimeter of 288 meters and an area of 4,860 square meters. What are the dimensions of the field?

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Question 1193348: A soccer field has a perimeter of 288 meters and an area of 4,860 square meters. What are the dimensions of the field?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

L = length
W = width

P = perimeter of a rectangle
P = 2(L+W)
2(L+W) = P
2(L+W) = 288
L+W = 288/2
L+W = 144
L = 144-W

area of a rectangle = length*width
A = L*W
L*W = A
(144-W)*W = 4860
144W - W^2 = 4860
0 = W^2 - 144W + 4860
W^2 - 144W + 4860 = 0

Use the quadratic formula to solve for W
Plug in a = 1, b = -144, c = 4860
W+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

W+=+%28-%28-144%29%2B-sqrt%28%28-144%29%5E2-4%281%29%284860%29%29%29%2F%282%281%29%29

W+=+%28144%2B-sqrt%281296%29%29%2F%282%29

W+=+%28144%2B-+++36%29%2F%282%29

W+=+%28144%2B36%29%2F%282%29 or W+=+%28144-36%29%2F%282%29

W+=+%28180%29%2F%282%29 or W+=+%28108%29%2F%282%29

W+=+90 or W+=+54

If W = 90, then L = 144-W = 144-90 = 54
If W = 54, then L = 144-W = 144-54 = 90
Either way we get the values 54 and 90 in either order.

This shows that the order of L and W doesn't matter.
Typically L is larger than W, but it's simply a style choice than anything.

Answer: The rectangular field is 90 meters by 54 meters

Check:
area = L*W = 90*54 = 4860 square meters
perimeter = 2(L+W) = 2*(90+54) = 288 meters
The answer is fully confirmed.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
A soccer field has a perimeter of 288 meters and an area of 4,860 square meters.
What are the dimensions of the field?
~~~~~~~~~~~~~~~~~ 

Let x be the length, in meters;  y be the width.


Then x + y = 288/2 = 144  (half of the perimeter).


Consider the arithmetic mean of x and y, which is  %28x%2By%29%2F2 = 144%2F2 = 72.


Then, obviously,  x = 72+a, y = 72-a  for some value of "a"  (x and y are equally remoted from the arithmetic mean).


Therefore, the area of the field is

    area = x*y = (72+a)*(72-a) = 72%5E2 - a%5E2 = 5184 - a%5E2.


So, the area equation is  

    5184 - a%5E2 = 4860.


From this equation,  find  "a"

    a%5E2 = 5184 - 4860

    a%5E2 =     324

    a   =  sqrt%28324%29 = 18.


Thus the dimensions of the field are  x = 72+18 = 90 meters  and  y = 72-18 = 54 meters.


ANSWER.  The dimensions of the field are  90 meters by 54 meters.

Solved.

The method showed in my post is non-standard method,  which is rarely used in traditional school education.

Its advantage is in that in many cases it allows to get the answer without long calculations -
practically  MENTALLY,  as you see in the given case.

----------------

If you want to see many other similar solved problems,  look into the lessons
    - Problems on the area and the dimensions of a rectangle
    - Three methods to find the dimensions of a rectangle when its perimeter and the area are given
in this site.

The first reference is the standard method;  the second reference teaches non-standard methods.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Dimensions and the area of rectangles and circles and their elements".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.