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aesmithswhs
12.02.2020 •
Mathematics
When a vertical beam of light passes through a transparent medium,the rate at which its intensity I decreases is proportional toI(t), where t represents the thickness of the medium (infeet). In clear seawater, the intensity 3 feet below thesurface is 25% of the initial intensity I0 of theincident beam.
What is the intensity of the beam 15 feet below the surface?
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Ответ:
The intensity of the beam at 15feet below the surface is 0.098%Io
Step-by-step explanation:
The differential equation for the intensity of beam of light is given by: dI/dt= KI
dI/I = Kdt
Integrating both sides
Integral dI/I = integral Kdt
Ln/I/ = Kt + c1
I = e^(Kt + c1)
I= e^ it e^c1
I = Ce^it
At I(0) = Io
Io = C = I = Io e^ kt
At 3ft below surface
0.25 Io = Io e^3k
0.25 = e^3k
3k = Ln(0.25)
K = Ln(0.25)/3
K= 0.4621
I = Ioe^(-0.4621)
At 15 ft
I = Ioe^(-0.4621 ×15)
I= Io e^-0.69315
I = 0.00098Io
I= 0.098% Io
Ответ: