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 13 feet below the surface? (Give your answer in terms of I₀. Round any constants or coefficients to five decimal places.)

Answer 1

Given, the rate of decrease of Intensity with the thickness of the transparent medium(t) is directly proportional to its intensity. Therefore:

Here, k is the proportionality constant, I is the intensity and I(t) is the intensity function with respect to thickness of the transparent medium (t).

Solving the above differential equation:

Integrating both sides:

where a is the constant of integration.

where C=e^a, a constant.

We have the initial condition, I(0)=I₀

Hence, the equation becomes:

We also know that at t = 3 feet, I=0.25I₀

Therefore, the final equation becomes:

When the depth is 13 feet, the intensity will be(on simplification):

Summary:

The Intensity of light beam at 13 feel below the surface is 0.00246 time initial intensity on the surface,