# Differentiation problem

1. Oct 3, 2004

### Monsu

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.
thanks!
Monsurat.

2. Oct 3, 2004

### Tide

Just write $\sin \theta$ in terms of D and R (radius of the circle) and set the derivative of I with respect to D equal to 0, etc.

3. Oct 3, 2004

$$\textrm{Here goes some hints...}$$

$$R = 20 \textrm{ m}$$

$$\hline$$

$$R = D \cos \theta$$
$$h = D \sin \theta = \frac{R}{\cos \theta} \cdot \sin \theta = R \tan \theta$$

$$\hline$$

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

$$\textrm{Good luck!}$$