```Question 230375
We need 2 Equations with 2 Variables.

Let w=width of the field
Let L=length of the field

Eqn 1: w*L=12000
Eqn 2: w+L-60=sqrt(w^2+L^2)

Note that in Eqn 2 I used pythag's theorem on the right-hand side to find the diagonal from one corner of the field to the other.

Isolate L in Eqn 1: L=12000/w and sub into Eqn 2:

w+(12000/w)-60=sqrt(w^2+(12000/w)^2)

This is a long equation with a lot of algebra. Square both sides of the equation to remove the radical:

(w+(12000/w)-60)(w+(12000/w)-60)=(w^2+(12000/w)^2)

w^2 +12000-60w+12000+(12000/w^2)^2 -(720000/w) -60w -(720000/w) +3600 = (w^2+(12000/w)^2)
w^2 - 120w - (1440000/w)+27600 = w^2 + (12000/w)^2 - (120000/w)^2
w^2 - 120w - (1440000/w)+27600 = w^2
-120w - (1440000/w)+27600 = w^2 - w^2
-120w - (1440000/w)+27600 = 0

Now multiply everything by w to get rid of the denominator under 1440000.
-120w^2 -1440000+27600w = 0

Now divide everything by -1:
120w^2 -27600w + 1440000 = 0

Now divide everything by 120:
w^2 -230w +12000 = 0

Now use the quadratic formula to solve for w, where A=1, B=-230, and C=12000.

Using the quadratic formula to solve, w=150 or 80.

Therefore, w=80, L=150.

The diagonal can be found by using Pythag's theorem:

Diagonal = sqrt(150^2+80^2)
Diagonal = 170

170 is 60 less than 230 (or 150+80, which is what Kim would walk if she walked along the edges).```