SOLUTION: A staphylococcus bacterial culture grows according to the formula P=P(sub o)a^t. if it takes 5 days for the culture to triple in size, how long will it take to double in size?

Question 622882: A staphylococcus bacterial culture grows according to the formula P=P(sub o)a^t. if it takes 5 days for the culture to triple in size, how long will it take to double in size?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First of all, let's make sure we understand the equation. The formula you were given, , is a general formula for exponential growth (or decay). The "P" is the population (or number) of bacteria after "t" units of time. The "" is P (the population) at time zero (i.e. the population of bacteria at the time we start measuring it). The "a" is a positive number that represents how fast P changes. (If a > 1 then this is an equation for exponential growth. If 0 < a < 1 then this is an equation for exponential decay.)

To solve this problem we will need to find the specific equation that applies. In other words we need to find the value for "a".

To find "a" we will use the fact that it takes 5 days for the bacteria to triple in number. If we start with bacteria, how many would we have after the bacteria triple? Triple means three times so if we start with bacteria then we would have after the bacteria triple. So, after replacing the P with and the t with 5 (since it takes 5 days to triple) we get:

We can solve this for a. Dividing by we get:

Fifth root of each side:

This makes the specific equation for this exponential growth problem:

Now can use this to find the time will take to double. Double means two times so and our equation is now:

Now we solve for "t". Dividing both sides by :

With a variable in an exponent like this, we usually use logarithms. Finding the ln of each side (log works, too):

Next we use a property of logarithms, , to "move" the exponent in the argument out in front. (It is this very property that is the reason we use logarithms. It allows us to move the exponent, where the variable is, to a place where we can "get at it" with algebra and solve for it.) Using this property we get:

Now we just divide by :

This is an odd but exact expression for how long it will take for the bacteria to double. If you want a "normal" number, then get out your calculator. Note:

The ln's can be log's instead. The answer works out the same either way.
To find on your calculator, use that fact that roots are fractional exponents. For a fifth root the fraction would be 1/5. So . If you calculator has buttons for parentheses then you could just enter "3^(1/5)". Or, since 1/5 = 0.2, you could enter "3^0.2"

You should get something close to 3.15464877. So it will take approximately 3.15464877 days for the bacteria to double.
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