Chapter 4, Problem 4P

The Beer-Lambert Law As sunlight passes through the waters of lakes and oceans, thelight isabsorbed, and the deeper it penetrates, the more its intensity diminishes. The light intensityI at depth x is given by the Beer-Lambert Law:
$I=I_0{e}^{-kx}$
Where $I_0$ the light intensity at the surface and k isis 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.
(a) 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 kfor 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 $b^x=e^{In(b^x)}=e^{xInb}$ . See Example 1(b).)
(b) Make a scatter plot of the data, and graph the function that you found in part (a) on your scatter plot.
(c) If the 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.