Question from Amelia:
Dear Dr. Callahan, I found a place where the answer key for even-numbered problems in Algebra II with Trig is incomplete. In Ch 5 Section 5, problem 90 explains what a photic zone is in lakes or the ocean, and gives the formula I = Io e ^ -kd where k is the constant called the coefficient of extinction and d is the depth. It then gives the depth where, in a certain lake, half of the surface light remains, and asks you to find both k and the depth of the photic zone. However, the answer key only has the answer for the value of k. It says nothing at all about the depth of the photic zone.
Answer from Dr. Callahan:
Thanks for pointing this out. Actually there are TWO errors in this problem – so let me resolve it
and solving by taking the natural log (ln) of both sides we get
-0.693 = -k * 14.3
k = 0.0485 (In the Solutions to Selected Problems this is 0.485. If you tried with the other number 0.485, your answer would not make since because you get 1% of the light at about 9 feet which is more shallow than when you get half the light at 14 feet)
Now, to complete this you would need to take the k value and insert it in the equation where you are at 1% of surface light remaining.
This gives you
now take the natural log (ln) to get
-4.605 = -0.0485d
d = 94.95 feet
and rounding to 3 significant digits (See page A-3 in back of book)
d = 95.0 feet
CHECK OUR WORK
Now we can check the work!
I = Io e^(-0.0485*95)
I/Io = 0.00997 or 0.01 or 1%
Hope this clears it up.
PS – Sorry for the delay – I just thought you were telling us we were missing something and did not look at it close till today!