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pre-calc

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construction of a rain gutter: a piece of rectangular sheet metal is 20 inches wide. it is to be made into a rain gutter by turning the edges to form parallel sides. let x represent the length of each of the parallel.

a) give the restrictions on x.
b) determine a function of A that gives the area of a cross section of the gutter.
c) for what values of x will A be a maximum (and thus maximize the amount of the water that the gutter will hold)?
d) what is this maximum area?
e) for what values of x will the area of a cross section be less than 40 square inches?

  • pre-calc - ,

    Look at a cross-section of the gutter.
    let the height be x
    then the base width = 20-2x , assuming the gutter is open at the top

    a) in 20-2x , 0 < x < 10
    b) Area = x(20-2x)
    c) Area = 20x - 2x^2
    = -2(x^2 - 10x + 25 - 25 )
    = -2( (x-5)^2 - 25)
    = -2(x-5)^2 + 50
    d) maximum area is 50 when x = 5

    e) x(20-2x) < 40
    -2x^2 + 20x < 40
    x^2 - 10x + 20 > 0
    consider x^2 - 10x + 20 = 0
    x = (10 ± √20)/2
    = 5 ± √5

    so the area is less than 40 for
    5-√5 < x < 5+√5

  • pre-calc - ,

    thank you! that helped a lot

  • pre-calc - ,

    how did you figure c?

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