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Thread: Expression of Formula In Some Terms

  1. #1
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    Expression of Formula In Some Terms

    The formula for the area of a spherical cap with radius (the sphere's radius) $\displaystyle R$ and height $\displaystyle h$ is given as

    $\displaystyle A = 2{\pi}Rh$

    This formula can be expressed in terms of $\displaystyle R$ and $\displaystyle r$, where $\displaystyle r$ is the radius of the base of the spherical cap:

    $\displaystyle (R-h)^2 + r^2 = R^2$
    $\displaystyle (R-h)^2 = R^2 - r^2$
    $\displaystyle R-h = \sqrt{R^2 - r^2}$
    $\displaystyle h = R - \sqrt{R^2 - r^2}$

    $\displaystyle A = 2{\pi}R(R - \sqrt{R^2 - r^2})$

    Now, how do I express the formula in terms of $\displaystyle R^2$ and $\displaystyle \frac{r}{R}$? I've been trying for quite some time but I'm stuck...
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  2. #2
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    Quote Originally Posted by fishcake View Post

    This formula can be expressed in terms of $\displaystyle R$ and $\displaystyle r$, where $\displaystyle r$ is the radius of the base of the spherical cap:

    $\displaystyle (R-h)^2 + r^2 = R^2$
    $\displaystyle (R-h)^2 = R^2 - r^2$
    For $\displaystyle r$:

    $\displaystyle (R-h)^2 = R^2 - r^2$

    $\displaystyle R^2-2Rh+h^2 = R^2 - r^2$

    $\displaystyle -2Rh+h^2 = - r^2$

    $\displaystyle 2Rh-h^2 = r^2$

    $\displaystyle \sqrt{2Rh-h^2} = r$




    For $\displaystyle R$:

    $\displaystyle (R-h)^2 = R^2 - r^2$

    $\displaystyle R^2-2Rh+h^2 = R^2 - r^2$

    $\displaystyle -2Rh+h^2 = - r^2$

    $\displaystyle 2Rh-h^2 = r^2$

    $\displaystyle 2Rh = r^2+h^2$

    $\displaystyle R = \frac{ r^2+h^2}{2h}$
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  3. #3
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    Quote Originally Posted by fishcake View Post

    Now, how do I express the formula in terms of $\displaystyle R^2$ and $\displaystyle \frac{r}{R}$? I've been trying for quite some time but I'm stuck...


    For $\displaystyle R^2$:

    $\displaystyle
    R = \frac{ r^2+h^2}{2h} \implies R^2 = \left(\frac{ r^2+h^2}{2h}\right)^2 =\dots
    $




    For $\displaystyle \frac{r}{R}$:

    $\displaystyle \frac{r}{R} = \frac{\sqrt{2Rh-h^2}}{\frac{ r^2+h^2}{2h}} = \dots$
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  4. #4
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    Hmm, I'm still confused here. How do I make use of $\displaystyle \frac{r}{R}$ and $\displaystyle R^2$ inside the formula for $\displaystyle A$? What does "express $\displaystyle A$ in terms of $\displaystyle \frac{r}{R}$ and $\displaystyle R^2$" actually means anyway? Maybe I have misinterpreted the question, which can be found here. It's question 1(b).
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