finding the area of a hole of rectangle?

Question

a rectangular sheet of metal has dimensions of 250mm x 200mm. A rectangular hole with a distance "x" between it's edges and the edge of the plate is cut into the plate. the effect of adding the hole is to reduce weight of the plate by 60%.

(i) show that the area of the hole can be expressed as 4xsquared - 900x + 50000 = area of hole

(ii) solve the equation for "x" to determine the dimensions for the hole.

(iii) given that the area of the plate is 20000msquared clearly show steps and explain methodology at each point.

Answer 1

The area of the sheet is 250*200=50000 cm^2,
and area of the hole is 0.6*50000 = 30000 cm^2.
Area of the hole is
=(250-2*x)*(200-2*x)
=50000-500*X-400*X+4*X^2
=4*X^2-900*X+50000.
Area of the hole is= 30000 cm^2.
EQUATING above two, we have
4*X^2-900*X+50000=30000
4*X^2-900*X+20000=0
X^2-225*X+12500=0
x=-b+/-sqrt(b^2-4*a*c)/2*a
x=225+/-sqrt(225^2-4*1*12500)/2*1
x=225+/-sqrt(50625-50000)/2
x=225+/-sqrt(625)/2
x=225+/-(25)/2
x=(225+25)/2 or x=225-25/2
x=125 or x=100

Answer 2

It's just algebra. The area of the sheet is 250*200=50000 cm^2, and of the hole is 0.6*50000 = 30000 cm^2. The sides of the hole are 250-2*x and 200-2*x. So the area of the hole is (250-2*x)*(200-2*x) = 4*x^2 - 900*x + 50000 = 30000, or in standard quadratic form, 4*x^2 - 900*x + 20000 = 0.
You should be able to solve the quadratic; if not, google "quadratic equation" and you can learn all about it.
EDIT: Or just read on. In a spurt of originality, another answerer has saved you the trouble. Except (whoops!) namrata's roots are, at this moment, wrong. The roots are actually 200 and 25, of which only x=25 works with the problem definition.
Just to check, (250-2*25)*(200-2*25) = 200*150 = 30000.