The area A, in square meters, of a rectangle with a perimeter of 160 meters is given by the equation A = 80w − w2, where w is the width of t

Question

The area A, in square meters, of a rectangle with a perimeter of 160 meters is given by the equation A = 80w − w2, where w is the width of the rectangle in meters. What is the width of a rectangle if its area is 700 m2?

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Alaia 1 week 2021-09-15T21:28:17+00:00 1 Answer 0

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    2021-09-15T21:29:17+00:00

    Answer:

    the width is 10 m

    Step-by-step explanation:

    if the relationship between area and width is

    A = 80*w − w²

    for an area A=700 m² , we have

    700 m² = 80*w − w²

    w² – 80*w + 700 m² = 0

    aw² + b*w + c = 0

    where a=1 , b=-80 and c=700

    this quadratic equation has as solution the following formula

    w = [-b ± √ ( b² – 4*a*c) ]/(2*a)

    replacing values

    w = [80 ± √ ( 80² – 4*1*700) ]/(2*1) = (80 ± 60)/2

    then

    w₁=(80 – 60)/2 = 10 m

    w₂ =(80 + 60)/2 = 70 m

    since the area has the form A= length * width = 80*w − w² = (80− w)*w

    then the length of the rectangle is

    length = 80− w

    for w₁=10 m → length = 80− 10 = 70 m

    for w₁=70 m → length = 80− 70 = 10 m

    by definition the shorter side is the width ( and the longer one , the length) , therefore the only possible option is the first one .

    Thus the width is 10 m

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