(Solved): When a vertical beam of light passes through a transparent medium, the rate at which its intensity ...

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 \( I_{0} \) of the incident beam. Find the constant of proportionality \( k \), where \( \frac{d I}{d t}=k I \). \[ k= \] What is the intensity of the beam 10 feet below the surface? (Give your answer in terms of \( I_{0} \). Round any constants or coefficients to five decimal places.)