Question

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 14 feet below the surface

Answer 1

The intensity of the beam 14 feet below the surface is 0.155% of the initial intensity.

Step-by-step explanation:

Let the initial intensity be I₀

The intensity of the beam at a depth 3 ft below the surface of the water is 25% of I₀ = 0.25I₀

The differential equation for this relation is

dI/dx = kI

dI/I = kdx

∫(dI/I) = k ∫dx

In I = kx + c

I = eᵏˣ.eᶜ

At x = 0, I = I₀

I₀ = eᶜ

I = I₀ eᵏˣ

At x=3, I = 0.25 I₀

0.25I₀ = I₀ e³ᵏ

e³ᵏ = 0.25

3k = In 0.25 = -1.386

k = - 0.462

At x=14

I = I₀ eᵏˣ

kx = - 0.462 × 14 = - 6.47

I = I₀ e⁻⁶•⁴⁷

I = 0.00155 I₀

That is 0.155% of the initial intensity at the surface.