differential equations

Question

When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t),

 where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity I₀

 of the incident beam. What is the intensity of the beam 12 feet below the surface? (Give your answer in terms of I₀.

 Round any constants or coefficients to five decimal places.)

______________

Answer 1

Hi Destiny,

The rate at which the intensity I decreases is proportional to I(t), and we can represent this as the differential equation dI/dt = -kI, where k is the constant of proportionality and t is the thickness of the medium in feet.

First, separate variables and integrate:

Integrate (1/I) dI = -k integrate dt

Integration gives:

ln|I| = -kt + C

To find the constant C, use the initial condition I(0) = I0:

ln|I0| = -k * 0 + C, so C = ln|I0|

Thus, the general solution becomes:

ln|I| = -kt + ln|I0|

To find k, use the information that 3 feet below the surface, the intensity is 25% of I0:

ln(0.25I0) = -3k + ln(I0), so -3k = ln(0.25), which means k = -ln(0.25)/3

Finally, to find the intensity 12 feet below the surface, use t = 12 in the equation I = I0 * e^(-kt):

I = I0 * e^(-12k)

Substitute k = -ln(0.25)/3 to find I.

After calculation, I turns out to be I = I0 * (0.25)^4.

So, the intensity of the beam 12 feet below the surface will be I0 * (0.25)^4.

Thank you,

Benjamin M.

p.s. If you found this answer helpful, I would greatly appreciate any reviews or suggestions as I am new to the platform. :)