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Math Help - Differentials to calculate Maximum Error

  1. #1
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    Differentials to calculate Maximum Error

    The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm. Use differentials to estimate the maximum error in the calculated volume.
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  2. #2
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    Hello, drahcirnaw!

    I had to snake my way through this one.
    Hope I got it right . . .


    The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm.
    Use differentials to estimate the maximum error in the calculated volume.

    We have: . C \:=\:2\pi r

    We are given: . C = 90 \quad\Rightarrow\quad 90 \:=\:2\pi r \quad\Rightarrow\quad r \:=\:\frac{45}{\pi} .[1]


    We have: . dC \:=\:2\pi dr

    We are given: . dC \,=\,\pm0.5 \quad\Rightarrow\quad \pm 0.5 \:=\:2\pi dr \quad\Rightarrow\quad dr \:=\:\frac{\pm0.5}{2\pi} .[2]



    The volume of a sphere is: . V \:=\:\frac{4}{3}\pi r^3

    . . Then: . dV \;=\;4\pi r^2dr

    Substitute [1] and [2]: . dV \;=\;4\pi\left(\frac{45}{\pi}\right)^2\left(\frac{  \pm0.5}{2\pi}\right) \;=\;\pm\frac{2025}{\pi^2}


    Therefore, the maximum error in volume is about: . \pm205.2\text{ cm}^3

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  3. #3
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    Thank you so much Soroban. Makes much more sense now.
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