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Math Help - [SOLVED] Applications of Differentiation Question

  1. #1
    Member looi76's Avatar
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    [SOLVED] Applications of Differentiation Question

    Question:
    A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is 192\pi cm^2. The cylinder has a radius of r cm and a height of h cm.

    (i) Express h in terms of r and show that the volume, V cm^3, of the cylinder is given by:
    V = \frac{1}{2}\pi(192r - r^3)

    Given that r can vary,
    (ii) Find the value of r for which V has a stationary value,
    (iii) find this stationary value and determine whether it is a maximum or a minimum.


    (i) Can someone please show me how to get V = \frac{1}{2}\pi(192r - r^3)?
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  2. #2
    Member Danshader's Avatar
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    external surface area of a closed cylinder
    = 2\pi r h + 2\pi r^2

    external surface area for one open end of cylinder
    = 2\pi r h + \pi r^2
    = 192\pi

    obtain an expression for h.

    volume of cylinder
    = \pi r^2 h

    substitute the value of h into the volume equation
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  3. #3
    Member looi76's Avatar
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    Quote Originally Posted by Danshader View Post
    external surface area of a closed cylinder
    = 2\pi r h + 2\pi r^2

    external surface area for one open end of cylinder
    = 2\pi r h + \pi r^2
    = 192\pi

    obtain an expression for h.

    volume of cylinder
    = \pi r^2 h

    substitute the value of h into the volume equation
    Thanks Danshader

    2\pi{rh} + \pi{r^2} = 192\pi

    \frac{2\pi{rh}}{\pi} + \frac{\pi{r^2}}{\pi} = \frac{192\pi}{\pi}

    2rh + r^2 = 192

    2rh = 192 - r^2

    h = \frac{192 - r^3}{2r}

    = \pi{r^2h}

    =\pi{r^2}\frac{192 - r^3}{2r}

    V = \frac{1}{2}\pi{}(192r - r^3)

    Now how do I find the value of r?
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  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
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    Definition:A point 'u' at which the derivative of a function f(x) vanishes,i.e. f'(u) = 0 is called a stationary point.


    This means you have to find values of r such that V'(r) = 0.
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  5. #5
    Member Danshader's Avatar
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    yup just like what isomorphism said, a stationary point can be found when the derivative of V with respect to r is 0

    \frac{dV}{dr} = 0

    earlier you already prove the equation of the volume:

    V = \frac{1}{2}\pi{}(192r - r^3)


    hence all you need to do is to differentiate this equation with respect to r and equate it to 0:

    \frac{d}{dr}{(\frac{1}{2}\pi{}(192r - r^3)}) = 0

    solve for r. good luck
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