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Calculus/MA 166; Calc Two

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Expert: - 7/3/2010


Question
According to the Beer-Lambert Law, the light intensity at a depth of x meters below the surface of he ocean is I(x)=I (sub 0) a^x, where I (sub 0) is the light intensity at the surface and a is a constant such that 0<a<1.
a) Express the rate of change of I(x) with respect to x in terms I(x).
b) If I(sub 0) = 8 and a = 0.38 find the rate of change of intensity with respect to depth at a depth of 20m.
c)Using the values form part b , find the average light intensity between the surface and a depth of 20m.

Thank you :)

Answer
a) The function is I(x) = I(0)*a^x, so the derivative is dI/dx = I(0)*ln(a)a^x
since I(0) is a constant.  The derivative is the rate of change.

b) Here, I(0) = 8 and a = 0.38, these can be put in and I(x) can be found.
To find dI/dx at 20 cm depth, since x is in meters, take x = 0.02 and put into the equation.

c) This requires finding ∫ x*I(x) dx from 0 to 20.
This is a u-v problem, with u = x and dv = I(x)dx.
From this, du = dx and v = I(0)*a^x/ln(a).

Put these values in and evaluate from x=0 to x=0.02.

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