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Math Help - Volume and Surface area derivative problem

  1. #1
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    Volume and Surface area derivative problem

    Hi I was hoping for some help on this 3 step problem:

    a) Given Surface area of sphere, A = 4*pi*r^2 and Volume = 4/3*pi* r^3, solve for V in terms of A.

    b) Find dV/dA.

    c) The volume of a ball of radius 4in is increased by .5 in^3. Estimate the increase in the
    area.

    I get stuck at part c...

    a) I solved for r in area formula and substituted in the volume formula to get:
    V= A/3* sqrt(A/4*pi)

    b) dV/dA = Sqrt(x)/4*sqrt(pi) *thanks to wolfram alpha

    c) My first thought was to find volume of a ball with radius 4, add .5 then solve for the new radius. Then plug that into the surface area formula and see what I get. However I feel there must be another way using part A and B.

    Any help is appreciated. Thanks.
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  2. #2
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    Re: Volume and Surface area derivative problem

    V = \frac{1}{6\sqrt{\pi}} \cdot A^{\frac{3}{2}}

    \frac{dV}{dA} = \frac{1}{4\sqrt{\pi}} \cdot A^{\frac{1}{2}}

    when r = 4 , A = 64\pi

    \frac{dV}{dA} = \frac{1}{4\sqrt{\pi}} \cdot 8\sqrt{\pi} = 2

    dA = \frac{dV}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4} \, in^2
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