Calculus Optimization question. Help.?

Question

A light is suspended at a height h above the floor. The illumination at the point P is inversely proportional to the square of the distance from the point P to the light and directly proportional to the cosine of the angle theta. (a) How far from the floor should the light be to maximize the illumination at the point P? r = distance from Light to P. h = distance from light to floor below. There are 10 meters from point P to a point O (spot on floor directly below light) (b) State the function you need to maximize or minimize and what the interval of admissible values is for the independent variable. (c) Verify your answers by testing all relevent critical points and the behavior at the ends of the interval.

Answer 1

The distance from lamp to P is r = √[h^2+s^2], where s = dist from center to P (10m). The illumination is inversely proportional to r^2, and directly proportional to cos(theta), where theta is going to be arctan(s/h). So the formula is

I(h) = cos(arctan(s/h))/(r^2+s^2)

By drawing a right triangle with legs l and h, you can find that cos(arctan(l/h)) = h/√[h^2+s^2] so the formula is then

I(h) = h/(h^2+s^2)^(3/2) this is the function to maximize w.r.t. h, the independent variable.

To maximize, differentiate and set to zero, you will get h = s/√2, and you can check it by putting into the formula h = (s/√2)+∆ and h = (s/√2)-∆ where ∆ is a small deviation.

Answer 2

you need to define theta here.