SOLUTION: The water in Midwestern lake contains sediment, and the presence of the sediment reduces the transmission of light through the water. Experiments indicate that the intensity of lig

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The water in Midwestern lake contains sediment, and the presence of the sediment reduces the transmission of light through the water. Experiments indicate that the intensity of lig      Log On


   

Question 1057455: The water in Midwestern lake contains sediment, and the presence of the sediment reduces the transmission of light through the water. Experiments indicate that the intensity of light is reduced by 10% by passage through 20 cm of water. Suppose that the lake is uniform with respect to the amount of sediment contained by the water. A measuring instrument can detect light at the intensity of 0.17% of full sunlight. This measuring instrument is lowered into the lake. At what depth will it first cease to record the presence of light? Give your answer to the nearest 10 cm.
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
This is supposed to be done through either logarithms or exponentiation, but here is instead a crude way: 
Depth     Intensity
0          100%
20          90
40          81
60          72.9
80          65.61
100         59.05
120         53.14
140         47.8
160         43.05
180         38.74
200         34.9
220         31.38


Light intensity depends on depth.
I=100%2Ae%5E%28-kx%29, start for a model of exponential decay.
You only need two data points, the FIRST two, for finding k.

Take log of both sides.
ln%28I%29=ln%28100%2Ae%5E%28-kx%29%29
ln%28I%29=ln%28100%29-kx
highlight%28ln%28I%29=-kx%2Bln%28100%29%29---------this is a LINEAR equation in variables ln%28I%29 and x; those being natural log of the amount of sunlight intensity, and the depth in the water. Notice very carefully that -k is the slope; and you can take ANY TWO DATA POINTS that you want, for finding the slope.