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Math Help - Calculus:Derivatives:Differential (I am stuck on this question help PLEASE)

  1. #1
    nch
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    Calculus:Derivatives:Differential (I am stuck on this question help PLEASE)

    In a manufacturing process, ball bearings must be made with radius of 0.4 mm, with a maximum error in the radius of 0.013 mm. Estimate the maximum error in the volume of the ball bearing.
    Solution: The formula for the volume of the sphere is________. If an error ∆r is made in measuring the radius of the sphere, the maximum error in the volume is ∆V=__________.
    Rather than calculating ∆V, approximate ∆V with dV, where dV=__________.
    Replacing r with___ and dr=∆r with ____ gives dV= ____.
    The maximum error in the volume is about____mm^3.
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  2. #2
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    Hello, nch!

    Some of this is just arithmetic . . .


    In a manufacturing process, ball bearings must be made with r \,=\,0.4\text{ mm,}
    with a maximum error in the radius of: . \Delta r = dr = \pm0.013\text{ mm.}

    Estimate the maximum error in the volume of the ball bearing.

    Solution: The formula for the volume of the sphere is: . \boxed{V \:=\:\tfrac{4}{3}\pi r^3}

    If an error \Delta r is made in measuring the radius of the sphere,
    the maximum error in the volume is: . \Delta V \:=\:\boxed{ {\color{white}XXX} }
    The volume of the correct sphere is: . V_o \:=\:\tfrac{4}{3}\pi (0.4)^3 \:=\:\frac{0.256}{3}\pi\text{ mm}^3

    With an error of \Delta r = 0.013, the radius is: . r \:=\:0.4 + 0.013 \:=\:0.413\text{ mm.}
    And the volume is: . V_1 \;=\;\tfrac{4}{3}\pi(0.413)^3 \;\approx\;\frac{0.28178}{3}\pi\text{ mm}^3

    The error in volume is: . \Delta V \;=\;\frac{0.28178}{3}\pi - \frac{0.256}{3}\pi

    . . Hence: . \Delta V \;=\;0.00859\pi\text{ mm}^3




    Rather than calculating \Delta V, approximate \Delta V with dV, where: .  dV \:=\:\boxed{4\pi r^2dr}

    Replacing r with 0.4,and dr =\Delta R with \pm0.013 gives: . dV \:=\:\boxed{{\color{white}XX}}
    dV \;=\;4\pi(0.4)^2(0.13) \;=\;0.00832\pi\text{ mm}^3

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  3. #3
    nch
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    thankz a lot
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