Suppose a population of bacteria in a Petri dish has a doubling time of 6 hours. How long will it take for an initial population of 11000 bacteria to reach 20000? Round your answer to two decimal places.
Well, bacteria sure know how to party! With a doubling time of 6 hours, those little guys are multiplying like crazy. Let me grab my clown calculator and do some quick math for you.
To go from 11,000 bacteria to 20,000 bacteria, we need to calculate how many doubling cycles it would take.
The formula for calculating the number of doubling cycles is:
Number of doubling cycles = log2 (final population/initial population)
So, using this formula, we can calculate the number of doubling cycles:
Number of doubling cycles = log2 (20000/11000)
Now, to find the time it takes for these doubling cycles, we multiply it by the doubling time of 6 hours:
Total time = Number of doubling cycles * doubling time
Well, don't leave me hanging! Let me just use my hilarious clown calculator and give you the answer you've been waiting for...
*Dings, buzzes, and squeaks*
After numerous calculations and infinite clown laughter, the time it will take for the initial population of 11,000 bacteria to reach 20,000 is approximately:
15.68 hours!
So, to put it simply, in less than 16 hours, those bacteria will be throwing a thriving party for 20,000 attendees. Enjoy the show!
To find out how long it will take for the bacteria population to reach 20,000 from an initial population of 11,000, we can use the formula for exponential growth:
N(t) = N(0) * (2^(t/d))
Where:
N(t) = population at time t
N(0) = initial population
t = time
d = doubling time
Plugging in the given values:
20000 = 11000 * 2^(t/6)
Dividing both sides by 11000:
2^(t/6) = 20000/11000
Taking the natural logarithm of both sides:
ln(2^(t/6)) = ln(20000/11000)
Using the property of logarithms: ln(a^b) = b * ln(a)
(t/6) * ln(2) = ln(20000/11000)
Simplifying:
t/6 = ln(20000/11000) / ln(2)
Multiply both sides by 6:
t = 6 * (ln(20000/11000) / ln(2))
Calculating the value:
t = 6 * (3.172 / 0.6931)
t ≈ 6 * 4.57
t ≈ 27.42 hours
Therefore, it will take approximately 27.42 hours for the population to reach 20,000 bacteria.
To answer this question, we need to use the concept of exponential growth.
The doubling time of 6 hours means that the bacteria population doubles every 6 hours. Let's break down the problem:
1. Determine the growth rate: The growth rate can be calculated by dividing the doubling time by the natural logarithm of 2 (ln(2)). In this case, the growth rate would be (6 hours / ln(2)).
2. Use the growth rate to calculate the time it takes to reach the desired population: We can use the formula for exponential growth:
N(t) = N0 * e^(rt)
Where:
N(t) = final population size
N0 = initial population size
e = mathematical constant approximately equal to 2.71828
r = growth rate
t = time
We want to find the time (t) it takes for the population to reach 20000, given an initial population (N0) of 11000. We already know the growth rate (r). So, we can rearrange the formula and solve for t:
t = (ln(N(t) / N0)) / r
3. Plug in the values and calculate the time: Substitute the given values into the formula we derived in step 2. The final population (N(t)) is 20000, the initial population (N0) is 11000, and the growth rate (r) is (6 hours / ln(2)). Once you plug in these values, you can use a calculator to solve for t.
Let's calculate the time it takes for the population to reach 20000 bacteria:
t = (ln(20000 / 11000)) / (6 hours / ln(2))
Using a calculator, the value of t is approximately 4.79 hours (rounded to two decimal places).
Therefore, it will take approximately 4.79 hours for the initial population of 11000 bacteria to reach 20000 bacteria.
You want t where
2^(t/6) = 20000/11000 = 1.818
t = 6log1.818/log2 = 5.174 hr
Makes sense, since in 6 hrs it will double to 22,000