Chapter 4.FOM, Problem 4P

The Beer-Lambert Law As sunlight passes through the waters of lakes and oceans, the light is absorbed, and the deeper it penetrates, the more its intensity diminishes. The light intensity I at depth x is given by the Beer-Lambert Law:

$I=I_0{e}^{-kx}$

where $I_0$ is the light intensity at the surface and k is a constant that depends on the murkiness of the water (see page 402). A biologist uses a photometer to investigate light penetration in a northern lake, obtaining the data in the table.

Light intensity decreases exponentially with depth.

1. Use a graphing calculator to find an exponential function of the form given by the Beer-Lambert Law to model these data. What is the light intensity $I_0$ at the surface on this day, and what is the “murkiness” constant k for this lake? [Hint: If your calculator gives you a function of the form $I=a{b}^x$, convert this to the form you want using the identities $b^x=e^{\ln \left(b^x\right)}=e^{x\ln b}$. See Example 1(b).]

Make a scatter plot of the data, and graph the function that you found in part (a) on your scatter plot.

If the light intensity drops below 0.15 lumen (lm), a certain species of algae can’t survive because photosynthesis is impossible. Use your model from part (a) to determine the depth below which there is insufficient light to support this algae.

Depth (ft)	Light intensity (lm)	Depth (ft)	Light intensity (lm)
5 10 15 20	13.0 7.6 4.5 2.7	25 30 35 40	1.8 1.1 0.5 0.3