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Math Help - Volume of Spheres

  1. #1
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    Volume of Spheres

    The volume of a sphere of radius r is V = (4/3)(pi)(r^3); the surface area S of this sphere is S = 4(pi)(r^2).

    (a) Express the volume V as a function of the surface area S.

    (b) If the surface area doubles, how does the volume change?
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  2. #2
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    Quote Originally Posted by symmetry View Post
    The volume of a sphere of radius r is V = (4/3)(pi)(r^3); the surface area S of this sphere is S = 4(pi)(r^2).

    (a) Express the volume V as a function of the surface area S.

    (b) If the surface area doubles, how does the volume change?
    a) r = \left ( \frac{S}{4 \pi} \right )^{1/2}

    Thus:
    V = \frac{4}{3} \pi \left [ \left ( \frac{S}{4 \pi} \right )^{1/2} \right ]^3

    V = \frac{4 \pi}{3(4 \pi)^{3/2}} S^{3/2}

    V = \frac{1}{3(4 \pi)^{1/2}} S^{3/2}

    b) If S \to 2S then
    V \to \frac{1}{3(4 \pi)^{1/2}} (2S)^{3/2}

    V \to 2^{3/2} \left ( \frac{1}{3(4 \pi)^{1/2}} S^{3/2} \right )

    V \to 2^{3/2}V

    -Dan
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  3. #3
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    ok

    Dan,

    Hello. How did you get r to equal (s/4pi)^(1/2)?

    I did not follow that part.

    Thanks!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by symmetry View Post
    Dan,

    Hello. How did you get r to equal (s/4pi)^(1/2)?

    I did not follow that part.

    Thanks!
    S = 4 \pi r^2

    \frac{S}{4 \pi} = \frac{4 \pi r^2}{4 \pi}

    r^2 = \frac{S}{4 \pi}

    Now, we need to take the square root of both sides. It was more convenient for me to express this as
    r = \left ( \frac{S}{4 \pi} \right )^{1/2}
    rather than
    r = \sqrt{\frac{S}{4 \pi}}

    Both of these forms are completely equivalent, though you would probably recognize the second as being more familiar. Perhaps I should have mentioned this when I solved the problem.

    -Dan
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  5. #5
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    ok

    I finally got it.

    Thanks!
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