Question

Optimization?

A light source is located over the center of a circular table of diameter 4 feet. (See picture below) Find the height h of the light source such that the illumination I at the perimeter of the table is maximum when

`l = \frac{ksin \alpha }{ s^{2}}`

where s is the slant, `\alpha` is the angle at which the light strikes the table and k is a constant.

Answer 1

Write `I` in terms of `h`.

Explanation:

We are given that `I = (ksin alpha)/s^2`

Use the definition of `sin alpha` for acute angle `alpha`.

`sin alpha = h/s`

So `I = (ksin alpha)/s^2 = k sin alpha 1/s^2 = k(h/s)*1/s^2`

That is: `I = (kh)/s^3`

Now use the Pythagorean Theorem to get

`s = sqrt(4+h^2) = (4+h^2)^(1/2)`.

`I = (kh)/(4+h^2)^(3/2)`

Now maximize as usual. (Find and test the critical points for `I`.)

I get `h = sqrt2`