The area of the original lot is 18000 sq. ft.
Let l be the length of the original rectangle. Then l+30 is the width.
The length of the second rectangle is increased by 40; this would be l+40. The width of the second rectangle is increased by 20; this would be l+30+20=l+50.
The area of the second rectangle is found by multiplying its length and width; this gives us
(l+50)(l+40)=27200
Multiplying, we have
l*l+40*l+50*l+50*40=27200
l²+40l+50l+2000=27200
l²+90l+2000=27200
For quadratic equations, we want them equal to 0; subtract 27200 from both sides:
l²+90l-25200 = 0
We will use the quadratic formula to solve this:
[tex]l=\frac{-b\pm \sqrt{b^2-4ac}}{2a}
\\
\\=\frac{-90\pm \sqrt{90^2-4(1)(-25200)}}{2(1)}
\\
\\=\frac{-90\pm \sqrt{8100--100800}}{2}
\\
\\=\frac{-90\pm \sqrt{8100+100800}}{2}=\frac{-90\pm \sqrt{108900}}{2}
\\
\\=\frac{-90\pm 330}{2}=\frac{-90-330}{2}\text{ or }\frac{-90+330}{2}
\\
\\=\frac{-420}{2}\text{ or }\frac{240}{2}=-210\text{ or }120[/tex]
Since a negative length makes no sense, l=120. This means the width is l+30=150, and the area is 120(150) = 18000.
