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Homework Help: Differentiation problem

  1. Oct 3, 2004 #1
    hi, could anyone guide as to how to go about solving this question?
    A cicular patch of grass or radius 20m is surrounded by a walkway and a light is placed atop a lamppost at the circle's center. At what height shoud the light be placed to illuminate the walkway most strongly? The intensity of illumination "I" of a surface is given by
    I = [k.sin(theta)] / D^2 where is the distance from the light source to the surface and theta is the angle at which light strikes the surface, and k i s a positive constant

    pls anyone, just give me a push, i am competely cluess as to where to begin.
  2. jcsd
  3. Oct 3, 2004 #2


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    Just write [itex]\sin \theta[/itex] in terms of D and R (radius of the circle) and set the derivative of I with respect to D equal to 0, etc.
  4. Oct 3, 2004 #3
    [tex] \textrm{Here goes some hints...} [/tex]

    [tex] R = 20 \textrm{ m} [/tex]

    [tex] \hline [/tex]

    [tex] R = D \cos \theta [/tex]
    [tex] h = D \sin \theta = \frac{R}{\cos \theta} \cdot \sin \theta = R \tan \theta [/tex]

    [tex] \hline [/tex]

    [tex] I = k\cdot \frac{\sin \theta}{D^2} [/tex]
    [tex] \sin \theta = \frac{ID^2}{k} [/tex]
    [tex] \frac{h}{D} = \frac{ID^2}{k} [/tex]
    [tex] h = \frac{ID^3}{k} [/tex]
    [tex] h = \frac{I}{k}\left( \frac{R}{\cos \theta} \right)^3 [/tex]​

    [tex] \textrm{Good luck!} [/tex] :smile:
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