Math

A rectangular dog pen is to be constructed using a barn wall as one side and 60 meters of fencing for the other three sides. Find the dimensions of the pen that maximize the pen's area.

asked by booyah
  1. 20 x 20

    posted by LULU
  2. let the length be y m (the single side)
    let the width be x m

    so we know that 2x + y = 60
    or y = 60 - 2x

    Area = xy
    = x(60-2x)
    = -2x^2 + 60x

    At this point I don't if you know Calculus or not.
    If you do, then
    d(Area)/dx = -4x + 60
    = 0 for a max of area
    -4x + 60 = 0
    x = 15
    then y = 30

    if you don't take Calculus, we have to complete the square
    area = -2(x^2 - 30x)
    = -2(x^2 - 30x + 225 - 225)
    = -2((x-15)^2 - 225)
    = -2(x-15)^2 + 450

    so the area is a maximum when x = 15
    and then y = 60-30 = 30

    by either method,
    the width has to be 15 m, and the length has to be 30 m for a maximum area of 15(30) or 450 m^2

    (LULU's answer only gives an area of 400 )

    posted by Reiny

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