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Math Help - Sanity Check: Related Rates for Sphere

  1. #1
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    Sanity Check: Related Rates for Sphere

    Hello,

    Working on the following problem:

    Express the rate at which the volume of the sphere changes with respect to the surface area of the sphere (as a function of r).

    My Work:
    S = 4\pi r^2 \Rightarrow r = \left( \frac{S}{4\pi}\right)^{\frac{1}{2}}

    V = \frac{4}{3} \pi r^3 = \frac{4}{3}\pi \cdot \left( \frac{S}{4\pi}\right)^{\frac{3}{2}}

    \frac{d}{dS} \left( V = \frac{4}{3} \pi \cdot \left( \frac{S}{4\pi}\right)^{\frac{3}{2}\right)

    \frac{dV}{dS} = \frac{1}{2} \left(\frac{S}{4\pi}\right)^{\frac{1}{3}}

    \text{Substitute } S:
    \frac{dV}{dS} = \frac{1}{2} \left( \frac{4\pi r^2}{4\pi}\right)^{\frac{1}{3}} = \frac{1}{2} (r^2)^{\frac{1}{3}} = \frac{1}{2} \cdot r^{\frac{2}{3}}

    My brain is fried after doing a lot of calculus problems so I wanted a sanity check: does it look okay?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Sanity Check: Related Rates for Sphere

    You have made a slight error. You should have:

    \frac{dV}{dS}=\frac{1}{2}\left(\frac{S}{4\pi} \right)^{\frac{1}{2}}

    You could also use the chain rule as follows:

    \frac{dV}{dS}=\frac{dV}{dr}\cdot\frac{dr}{dS}=(4 \pi r^2)\left(\frac{1}{8\pi r} \right)=\frac{r}{2}
    Thanks from Biff
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