Problema Solution
Marine life is dependent upon the microscopic plant life that exists in the photic zone, a zone that goes to a depth where about 1% of the surface light still remains. Light intensity is reduced according to the exponential function:
I=Ioe^-kd
where I is the intensity d feet below the surface and Io is the intensity at the surface. The constant k is called the coefficient of extinction. At Crystal Lake in Wisconsin it was found that half the surface light remained at a depth of 14.3 feet. Find k, and find the depth of the photic zone. Compute answers to three significant digits.
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I=Ioe^-kd
I is the intensity
d feet below the surface
Io is the intensity at the surface
k is a constant called the coefficient of extinction
At Crystal Lake in Wisconsin we have
I = Io/2 at d = 14.3 ft
we need to find k
Io/2 = Io e^(-k*14.3) divide both sides by Io
1/2 = e^(-k*14.3)
e^(-k*14.3) = 1/2
ln e^(-k*14.3) = ln 0.5
-k*14.3 = ln 0.5
k = - (ln 0.5)/14.3
k = 0.0485
the coefficient of extinction at Crystal Lake in Wisconsin is k = 0.0485.