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19 June, 08:37

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.

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  1. 19 June, 12:00
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    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.
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