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Rectangular plot, find diagonals

Q: The area of a rectanglar plot is 12,000 square meters. If Kim walks along the diagonal from one corner to the opposiite corner instead of walking along the edges, she walks 60 meters less. What is the length of the diagonal? Thanks!

A: We need two Equations with two Variables. This is a very long and advanced question!

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

Eqn 1: w*L=12000
Eqn 2: w+L-60=√(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=√(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 = √(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).