The number of bacteria in a culture increases exponentially with a growth constant of .2 hour. How long will it take for the population to increase from 3000 to 25,000? Thank you for any help you can give and for your time!
To find the time it takes for the bacterial population to increase from 3000 to 25,000, we'll need to use the exponential growth formula:
N(t) = N0 * e^(kt)
Where:
N(t) is the population at time t
N0 is the initial population
e is the natural logarithm base (approximately 2.71828)
k is the growth constant
t is the time elapsed
In this case, we are given the growth constant as 0.2 hour, the initial population as 3000, and we want to find the time it takes for the population to reach 25,000.
So, we substitute the given values into the formula:
25000 = 3000 * e^(0.2t)
Now, we can solve for t:
e^(0.2t) = 25000 / 3000
To isolate the exponential term, we can take the natural logarithm (ln) of both sides:
ln(e^(0.2t)) = ln(25000 / 3000)
0.2t * ln(e) = ln(25000 / 3000)
Since ln(e) is equal to 1, we can simplify further:
0.2t = ln(25000 / 3000)
Now, we can solve for t by dividing both sides by 0.2:
t = ln(25000 / 3000) / 0.2
Using a calculator, we can find the natural logarithm of the expression 25000 / 3000 and divide the result by 0.2 to obtain the value of t. This will give us the time it takes for the population to increase from 3000 to 25,000.