Here is a problem I'm having trouble with:

50 8.0

3. Underwater light intensity decreases with increasing depth: the IDTRMn

deeper you go, the darker it gets. The relationship is approximately

an exponential attenuation, such that

A =B exp(-Kz)

where A is light intensity at depth z, B is light intensity at

depth 0, and K is the attenuation coefficient. The table below

lists depth z on the left and measurements of A on the right. Find K

A when z = 0 (which is the same thing as B).

2 1172.2

4 963.1

6 737.1

8 636.6

10 512.9

12 438.2

14 341.3

16 284.6

18 223.1

20 176.1

22 158.0

24 115.7

26 84.8

28 85.7

30 67.9

32 54.4

34 42.5

36 36.0

38 28.7

40 22.1

42 18.0

44 14.7

46 12.0

48 9.8

50 8.0

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Alright, so what I did to try and solve this was set up a system of equations. I solved for B at a depth of 10, and plugged in that equation for the constant B itself in a new depth equation for depth 12. This gave me an equation of 438.2 = (512.9/(e^-K(10)))*e^-K(12). I solved for K, got a number, then plugged that value back into the equation and solved for B.

Now the values I got using this method for B and K both worked for each of these equations. However, when I tried using the same values for different depths, the answers got further and further away from the actual values. I guessed that solving for two depths next to each other in the table was giving me a poor approximation, so what I did was I used the same method for depth 2 and depth 50. This gave me a better approximation, but I'm still off by a fair number the furthest away from either values for 2 or 50 (depth 30 is 67.9 but I got something like 64 or 63).

I could probably use the latter method here to get a relatively good answer for depth zero, but this is just bugging me. How do I solve this so that I can get the values of K and B to be almost exact algebraically? I might be able to plug this into a spreadsheet and do it that way, but that's not exactly showing my work or explaining how the answer can be obtained.