SOLUTION: If after 8 hr there are 8,262 bacteria remaining in a culture taht started with 6,000, how many would be present after 24 hr?

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Question 423391: If after 8 hr there are 8,262 bacteria remaining in a culture taht started with 6,000, how many would be present after 24 hr?
Found 2 solutions by jsmallt9, ewatrrr:
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
The assumption is that this is an exponential growth situation. And if it is an exponential growth then it the equation that models it must fit the pattern of exponential functions:
y+=+a%2Ab%5Ex

In this case the y will be the number of bacteria and the x will be the number of hours since measurements were started.

To answer the question we must first figure out the specific values for "a" and "b". Once we have these values we will be able to simply put a 24 in for the x and then use the equation to find the y when x = 24.

To find two unknowns, like a and b, we must have two equations. We can get these equations from the initial information we were given. When the problem tells us that there were 8262 bacteria after 8 hours, it is telling us that y = 8262 when x = 8. We can use this in the general equation:
8262+=+a%2Ab%5E8
And when the problem tells us that we started with 6000 bacteria, it is telling us that y = 6000 when x = 0. Putting these values into the general equation we get:
6000+=+a%2Ab%5E0
We now have the two equations we need to find a and b. The second equation, since b%5E0+=+1, simplifies to:
6000 = a
Already we have found one the the numbers we were looking for!. To find b we use the other equation and substitute the value we just found for a:
8262+=+6000%2Ab%5E8
Now we solve for b. We start with isolating the base and its exponent by dividing both sides by 6000:
8262%2F6000+=+b%5E8
Replacing the left side with its decimal form we get:
1.377+=+b%5E8
Now we find the 8th root of each side:
root%288%2C+1.377%29+=+root%288%2C+b%5E8%29
(Since the "b" in exponential equations is not allowed to be negative we can ignore the possible negative 8th roots of 1.277.) The right side simplifies:
root%288%2C+1.377%29+=+b
This is the exact value for b. If I find a decimal for this it will just be a close approximation for b. Because I prefer to deal with the exact numbers I am not going to bother finding the 8th root of 1.377. (If you want to do this then use your calculator to raise 1.377 to the 1/8 power. Since 1/8 as a decimal is 0.125 this would be: 1.377^0.125.)

Now that we have the a and the b we can write the specific exponential equation for this problem:
y+=+6000%2A%28root%288%2C+1.377%29%29%5Ex
And we can use this equation to find the number of bacteria after 24 hours. We just make x = 24:
y+=+6000%2A%28root%288%2C+1.377%29%29%5E24
At this point you might think that we need a decimal for the 8th root of 1.377. But we don't. Since an 8th root is the same as an exponent of 1/8 this expression will simplify without the decimal approximation:
y+=+6000%2A%28%281.377%29%5E%281%2F8%29%29%5E24
The rule for powers of a power is to multiply the exponemts:
y+=+6000%2A%281.377%5E%28%281%2F8%29%2A24%29%29
And since 1/8 times 24 is 3:
y+=+6000%2A1.377%5E3
Now we can cube 1.377 (which is not an approximation but the exact value for 8262/6000):
y = 6000*2.610969633
and multiply by 6000:
y = 15665.817798
As far as I know fractions of a bacteria are not possible. So we will round this off to the nearest whole number. So after 24 hours there should be about 15666 bacteria.

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

 P%5Bt%5D+=+P%5B0%5De%5Ert where r is the rate of decay and t the number of hours



 Question states***

   8262/6000 = e^8r

ln(8262/6000) = .3199 = 8r

     .3199 = 8r

       .04 = r

 P%5B24%5D+=+6000e%5E%28.04%2A24%29 = 15,670 present after 24 hours