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Math Help - Volume and Surface Area of a Sphere

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    Volume and Surface Area of a Sphere

    If the volume of the sphere doubles, what is the ratio of the surface area of the new, larger sphere to the old, smaller sphere? The answer is 2^(2/3). A good method for tackling these types of problems would be great. Should I just plug in numbers and work that way, or is there a more methodical way of doing it?
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    Re: Volume and Surface Area of a Sphere

    Write it down! The notation will help you.

    \frac{4}{3}\pi r_{1}^{3} = V_{1}

    \frac{4}{3}\pi r_{2}^{3} = V_{2}

    \frac{\frac{4}{3}\pi r_{2}^{3}}{\frac{4}{3}\pi r_{1}^{3}} = \frac{V_{2}}{V_{1}}

    \frac{r_{2}^{3}}{r_{1}^{3}} = \frac{V_{2}}{V_{1}}

    V_{2} = 2\cdot V_{1}

    \frac{r_{2}^{3}}{r_{1}^{3}} = \frac{2\cdot V_{2}}{V_{1}}

    \frac{r_{2}^{3}}{r_{1}^{3}} = 2

    Are we getting anywhere?

    The question is: \frac{4 \pi r_{2}^{2}}{4 \pi r_{1}^{2}} = What?
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    Re: Volume and Surface Area of a Sphere

    I'm still a little confused. I get steps 1-3, but I'm not sure exactly what you are doing in 4,5,6, and 7. Thanks for the help!
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    Re: Volume and Surface Area of a Sphere

    Hi benny92000,
    SA of sphere = pi d^2
    V of sphere = pi/6 * d^3
    let d1 =1 SA= pi V=pi/6
    V2 = 2* V1 = pi/3
    let X = the new diameter
    pi/6 *X^3=pi/3
    X^3 =2
    X= 2^1/3
    SA2 = pi*X^2=pi X^2 = pi * 2^2/3
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  5. #5
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    Re: Volume and Surface Area of a Sphere

    Quote Originally Posted by TKHunny View Post
    Write it down!
    ...
    V_{2} = 2\cdot V_{1}

    \frac{r_{2}^{3}}{r_{1}^{3}} = \frac{2\cdot V_{2}}{V_{1}} <-- small typo: Instead of V_2 it should be V_1

    ...
    Are we getting anywhere?

    The question is: \frac{4 \pi r_{2}^{2}}{4 \pi r_{1}^{2}} = What?
    Quote Originally Posted by benny92000 View Post
    I'm still a little confused. I get steps 1-3, but I'm not sure exactly what you are doing in 4,5,6, and 7. Thanks for the help!
    1. TKHunny showed you how to express the radius of the larger sphere ( r_2) by the radius of the smaller sphere:

    \frac{r_2^3}{r_1^3}=2~\implies~r_2 = \sqrt[3]{2} \cdot r_1

    2. Now calculate the ratio of the surfaces:

    \frac{4 \pi r_2^2}{4 \pi r_1^2}

    Sub in the term for r_2:

    \frac{4 \pi (\sqrt[3]{2} \cdot r_1)^2}{4 \pi r_1^2} = \frac{4 \pi \sqrt[3]{2^2} \cdot r_1^2}{4 \pi r_1^2}

    3. Cancel equal factors and re-write the cube-root term as a power.
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    Re: Volume and Surface Area of a Sphere

    Quote Originally Posted by earboth View Post
    small typo
    Gaa!!! You know how long I stared at that?! Anohter pair of eyes never hurts.
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