The perimeter of a rectangle is 200 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 900 square feet.

Length=x

width=y

Perimeter of a rectangle=2 (length) + 2 (width)

Therefore:

2x+2y=200

We simplify the equation dividend both sides of this equation by 2:

x+y=100

Then:

y=100-x

Area of a rectangle: length x width

We have the next inequation:

x (100-x) <900

100x-x²<900

x²-100x+900<0

We solve this inequation

1) we solve this equation:

x²-100x+900=0

x=[100⁺₋√ (10000-3600) ]/2 = (100⁺₋80) / 2

We have two solutions in this equation:

x₁=90

x₂=10

2) With these values, we make intervals:

(-∞,10)

(10,90)

(90,∞)

3) With these intervals, we check it out if the inequation works:

(-∞,10); for example; if x=0 ⇒ 0²-100 (0) + 900=100>0, this interval don't work.

(10,90); f. e: if x=11; ⇒ 11²-100 (11) + 900=-79<0, this interval works.

(90,∞); fe; if x=91 ⇒91² - (100) 91+900=81>0; this interval don't work.

answer: the possible lengths would be the values inside of this interval:

(10,90) ft.