A lake has been measured to have a ‘murkiness’ constant of .04.

Question

A lake has been measured to have a ‘murkiness’ constant of .04. If the intensity of light during the day is 20, how deep would you need to dive into the lake in order for the light’s intensity to drop to 12?

Answer 1

If the merkiness is a measure of how much light intensity drops per unit distance then we would be looking at the following equation.

dI/dy=-0.040I

dI/I=-.04dy

integrating both sides we get

ln(I)=-.04dy+c

Putting both sides to the as a power of e we get.

I=Ce^-.04y

^{Our original intensity at the surface is I}₀ so we get

I=I₀e^-0.04y

^{Now we are not interested in finding the intensity, we want to find y, so we solve for y.}

I=I₀e^-0.04y => I/I₀=e^-.04y => ln(I/I₀)=-0.04y => y= ln(I/I₀)/-0.04=ln(12/22)/-0.04=15.15

If you have any questions feel free to ask.