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

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

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.