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Math Help - Linear approximation

  1. #1
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    Linear approximation

    I need help with this problem, i dont know how to set it up.

    Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.02000 cm thick to a hemispherical dome with a diameter of 45.00 meters.
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    Quote Originally Posted by viet View Post
    I need help with this problem, i dont know how to set it up.

    Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.02000 cm thick to a hemispherical dome with a diameter of 45.00 meters.
    We have that volume of the hemisphere is a function of the radius, that is,
    V(r)=\frac{2}{3}\pi r^3
    Since .02 is a small number we have that the change in the derivative is approximately the change in the actual function, that is,
    dV\approx \Delta V
    Since,
    \frac{dV}{dr}=2\pi r^2
    The change in derivative is,
    dV=2\pi r^2 dr
    Where dr=\Delta r the change in the radius.
    Which is, +.02
    Thus,
    dV=2\pi (45)^2 (.02)
    Thus,
    dV=2\pi (2025)(.02)
    Thus,
    dV=81 \pi
    That means the change in volume is approximately increased by 81 \pi\mbox{ cm}
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