"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 16 feet below the surface

The intensity of the beam 16 feet below the surface is 0.06% of the initial intensity of the incident beam.

Step-by-step explanation:

"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).

This means that the intersity can be modeled by the following differential equation:

In which k is the decrease rate.

This differential equation leads to the following solution:

In which I(0) is the initial intensity.

The intensity 3 feet below the surface is 25% of the initial intensity I0 of the incident beam.

This means that I(3) = 0.25I(0). We use this to find k. So

So

What is the intensity of the beam 16 feet below the surface

This is I(16). So

The intensity of the beam 16 feet below the surface is 0.06% of the initial intensity of the incident beam.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

i pretty sure it is