When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional toI(t), where t represents the thickness of the medium (infeet). In clear seawater, the intensity 3 feet below thesurface is 25% of the initial intensity I0 of theincident beam.

What is the intensity of the beam 15 feet below the surface?

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

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for

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step-by-step explanation: