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differentiation problem |
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| Oct3-04, 02:01 AM | #1 |
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differentiation problem
hi, could anyone guide as to how to go about solving this question?
[B]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 [B] pls anyone, just give me a push, i am competely cluess as to where to begin. thanks! Monsurat. |
| Oct3-04, 02:46 AM | #2 |
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Recognitions:
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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.
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| Oct3-04, 02:47 AM | #3 |
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[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]
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