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Math Help - What maximum volume?

  1. #1
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    Question What maximum volume?

    USPS regulations require that any package must have length plus girth(distance around) of no more tha 84inches.Given this,what are the radius and height of a cylindrical package of maximum volume that is still mailable by parcel post?


    I know we need to maximise this.But am stuck.Any suggestion?
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  2. #2
    Senior Member
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    Dec 2008
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    We first state the problem in mathematical terms. Our formula for the volume of a cylinder is

    V=h\cdot\pi r^2.

    To maximize volume, we must have

    h+2\pi r=84

    or

    h=84-2\pi r.

    Substituting this into our formula for V, we have

    V=(84-2\pi r)\cdot \pi r^2.

    To find the maximum volume, we note that since V is differentiable everywhere and equals 0 at the endpoints of its domain, it must attain a maximum at a point where V'(r)=0.
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