Respuesta :
Answer:
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
Solving a differential equation, we get that the intensity of the beam 15 feet below the surface is 0.1% of the initial intensity.
-----------------------
The rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). This means that the situation is modeled by the following differential equation:
[tex]\frac{dI}{dt} = -kI[/tex]
In which k is the decay rate.
Applying separation of variables:
[tex]\frac{dI}{I} = -k dt[/tex]
[tex]\int \frac{dI}{I} = \int -k dt[/tex]
[tex]\ln{I} = -kt + C[/tex]
[tex]I(t) = Ce^{-kt}[/tex]
[tex]I(t) = I(0)e^{-kt}[/tex]
The intensity 3 feet below the surface is 25% of the initial intensity, thus I(3) = 0.25I(0), and we use this to find k.
[tex]I = I(0)e^{-kt}[/tex]
[tex]0.25I(0) = I(0)e^{-3k}[/tex]
[tex]e^{-3k} = 0.25[/tex]
[tex]\ln{e^{-3k}} = \ln{0.25}[/tex]
[tex]-3k = \ln{0.25}[/tex]
[tex]k = -\frac{\ln{0.25}}{3}[/tex]
[tex]k = 0.4621[/tex]
Thus:
[tex]I(t) = I(0)e^{-0.4621t}[/tex]
15 feet below the surface is I(15), thus:
[tex]I(t) = I(0)e^{-0.4621t}[/tex]
[tex]I(15) = I(0)e^{-0.4621(15)} = 0.001I(0)[/tex]
The intensity of the beam 15 feet below the surface is 0.1% of the initial intensity.
A similar problem is given at https://brainly.com/question/24206644
