? ? A light source is located over a circular table of diameter 4 feet. Find the

Question

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A light source is located over a circular table of diameter 4 feet. Find the height h of the light source such that the illumination l at the perimeter of the table is maximum when l = Ksin(teeta)/S2. S is the slant height. Teeta is the angle at which the light strikes the table, and kis a constant. ( see problem #40 in section 3.7 for a visuall of the problem). [HINT: Use your trigonometric knowledge to change las a function of alpha: i.e you should only have one variable and thatvariable should be teeta : note k is a number not a variable]??

Explanation / Answer

i = kh/s^3 correct

s = sqrt(h^2 + r^2) where r = diameter of table = 2 feet.

I = k h/(h^2 + 4)^3

Maxmimize h = take derivative and set = 0

d/dh of h(h^2 + 4)^-3/2

= (h^2 +4)^-3/2 +h(-3/2)(2h)(h^2 + 4)^-5/2 = 0

simplify into a fraction
then choose the numerator = 0 = 4 - 2h^2
2h^2 = 4
h^2 = 2
h = sqrt(2) ~ 1.4 ft

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