Question

after passing therough a material t centimeters thick, the intensity I(t) of a light beam is given by I(t) =(4^-ct)Io, where Io is the initial intensity and c is a constant called the absorption factor. Ocean water absorbs light with an absorption factor of c= .0101. at what depth will a beam of light be reduced to 50% of its initial intensity? 2% of its initial intensity?

Answer 1

I will do the last part
to have only 2% of the intensity left,
l(t) = .02

.02 = 4^(-.0101t) (1)
ln .02 = -.0101t ln4
-.0101t = ln .02 /ln4 = -2.821928
t = -2.821928/-.0101 = 279.4

for the 50% question , set the left side equal to .5