SOLUTION: please. could you explain in details for this college algebra problem? The population of a certain endangered species of owl is declining exponentially. There are currently

Question 424440: please. could you explain in details for this college algebra problem?

The population of a certain endangered species of owl is declining exponentially. There are currently 450 living specimens, whereas just 8 years ago there were 12,000 . If the population continues to decline exponentially, how long will it be until there is only a single owl left?

Found 2 solutions by nerdybill, Gersid:
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
The population of a certain endangered species of owl is declining exponentially. There are currently 450 living specimens, whereas just 8 years ago there were 12,000 . If the population continues to decline exponentially, how long will it be until there is only a single owl left?
.
apply exponential equation:
y = xe^(kt)
where
y is amount after time t
x is initial amount
k is a constant
t is time
.
plug in all given data to find k:
450 = 12000e^(8k)
450/12000 = e^(8k)
ln(450/12000) = 8k
ln(450/12000)/8 = k
-0.41043 = k
.
Our general equation is:
y = 12000e^(-0.41043t)
.
we now set y to 1 and solve for t:
1 = 12000e^(-0.41043t)
1/12000 = e^(-0.41043t)
ln(1/12000) = -0.41043t
ln(1/12000)/(-0.41043) = t
22.89 years = t

Answer by Gersid(33) (Show Source): You can put this solution on YOUR website!
This is a problem in exponential decay for which you can use the exponential function: , where the base, b, is between 0 and 1 for exponential decay. x is the number of years given in the problem (x = 8).
Start with the exponential function:
Substitute y = 450, the current number of owls. a = 12,000, the number of owls 8 years ago, and x = 8.
Solving for b will give you the exponential function for this situation.
Use your calculator to take the eighth root of both sides.
Approx. Now you can write the exponential function for this situation:

The question you want to answer is - For what value of x (x = number of years) will y (the number of owls remaining) = 1.
Solve for x. Divide both sides by 12000.
Take the logarithm of both sides.
From the power rule for logarithms.
Use your calculator to evaluate.
years.

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