Answer
(a) $I(x)lna$
(b) $-3.05\times10^{-8}$
(c) $0.41$
Work Step by Step
Given: $I=I_{0}a^{x}$
Where, $I_{0}$ is the light intensity at the surface and $a$ is a constant such that $0 < a < 1$.
The rate of change of with respect to x is given as follows:
$I'(x)=I_{0}a^{x}lna=I(x)lna$ ...(2)
(b) As we are given $I_{0}=8$ and $a=0.38$ , and our objective is to find the rate of change of intensity with respect to depth at a depth of 20 m.
$I(20)=8(0.38)^{20}\approx 3.15\times10^{-8}$
From equation 2, we have
The rate of intensity is given as:
$I'(20)=I(20)ln(0.38)= -3.05\times10^{-8}$
(c) The average light intensity between the surface and a depth of 20 m can be calculated as follows:
$I_{avg}(x)=\frac{I_{0}(a^{20}-1)}{20lna}$
As we are given $I_{0}=8$ and $a=0.38$
$I_{avg}(x)=\frac{8((0.38)^{20}-1)}{20ln(0.38)}=0.41$
