SOLUTION: A bacterial culture has an initial population of 10,000. If its population declines to 4000 in 8 hours what will it be at the end of 10 hours? Assume that the population decrease

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Question 281207: A bacterial culture has an initial population of 10,000. If its population declines to 4000 in 8 hours what will it be at the end of 10 hours? Assume that the population decreases according to the exponential model.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The general equation for exponential growth/decline is:
y+=+a%2Ab%5Ex
where "a" and "b" are positive constants.

For this problem of population decline we will use:
p+=+a%2Ab%5Et
where p is the population of bacteria and t is the time (in hours).

In order to find the population after 10 hours, we will need to find the "a" and "b" first. To find the "a" and "b" we will need two equations. The first equation will come from the fact that the initial population is 10000. Another way to say "initial population" is "population at t = 0". So the equation is:
10000+=+a%2Ab%5E0
Even tough we only know that "b" is positive, we know what b%5E0 is. Any non-zero number, like b, to the zero power is 1! This gives us:
10000+=+a%2A1
or
10000+=+a
So with one equation we have already found "a"! We can get a second equation from the fact that the population was 4000 after 8 hours. So (including the value we found earlier for "a"):
4000+=+10000%2Ab%5E8
We can solve this. Divide both sides by 10000:
4000%2F10000+=+b%5E8
which simplifies to:
0.4+=+b%5E8
Next we can find the 8th root of each side. (Since b must be positive we will not consider the negative 8th root of 0.4.)
root%288%2C+0.4%29+=+b
Now we finally have our equation:
p+=+10000%28root%288%2C+0.4%29%29%5Et

With this equation we can now find the population after 10 hours:
p+=+10000%28root%288%2C+0.4%29%29%5E10
This is the exact answer. But a decimal approximation will probably be more informative. You may already know how to use your calculator on the above expression.

But in case you don't, I'll rewrite it in a form that will make this easier.
First, an 8th root is an exponent of 1/8 so I'll replace the radical with an exponent of 1/8:
p+=+10000%28%280.4%29%5E%281%2F8%29%29%5E10
The rules for exponents say to multiply the 1/8 and the 10:
p+=+10000%28%280.4%29%5E%2810%2F8%29%29
The decimal for 10/8 is 1.25:
p+=+10000%28%280.4%29%5E%281.25%29%29
This form should be easy to put into your calculator. Raise 0.4 to the 1.25 power and then multiply that result by 10000. And finally, since fractions of bacteria do not exist (as far as I know), round this off to the nearest whole number. This will be the approximate population after 10 hours.