A culture of bacteria triples every 7 minutes. How long will it take a culture originally consisting of 24 bacteria to grow to a population of 100 000 bacteria? Show ALL work and round to one decimal place, if necessary.
Besides, we saw this one already. You just need to find x such that
24*3^(x/7) = 100000
If you posted the earlier version, with 77 minutes to triple, why did you not just make the fix and redo the calculation as shown? And don't bother to tell us to "show ALL work" ... Just ask your question. We help you find the solution. Following the instructions as required is your job.
A culture of bacteria triples after 7 hours.
What proportion of the original number of bacteria Q0 will be present after 14 hours?
What proportion of the original number of bacteria Q0 will be present after 28 hours?
To solve this problem, we can use the formula for exponential growth:
N = N0 * (3)^(t/7)
Where:
N = final population size (100,000 bacteria)
N0 = initial population size (24 bacteria)
t = time it takes for the population to grow to N
We are given that the population triples every 7 minutes, so the growth factor is 3.
Substituting the given values into the formula, we have:
100,000 = 24 * (3)^(t/7)
Now, we need to solve for t. Taking the natural logarithm (ln) of both sides of the equation will allow us to isolate the exponent:
ln(100,000) = ln(24 * (3)^(t/7))
Using the property of logarithms that ln(ab) = b * ln(a):
ln(100,000) = ln(24) + (t/7) * ln(3)
Next, we isolate the term containing t by subtracting ln(24) from both sides:
ln(100,000) - ln(24) = (t/7) * ln(3)
Using the property of logarithms that ln(a/b) = ln(a) - ln(b):
ln(100,000/24) = (t/7) * ln(3)
Now, we solve for t by multiplying both sides by (7 / ln(3)):
(t/7) * ln(3) = ln(100,000/24)
t/7 = ln(100,000/24) / ln(3)
t = 7 * (ln(100,000/24) / ln(3))
Using a calculator, we can find that ln(100,000/24) / ln(3) ≈ 13.8
Therefore, the time it will take for the population to grow to 100,000 bacteria is approximately 7 * 13.8 = 96.6 minutes. Rounded to one decimal place, this is approximately 96.6 minutes.
To find out how long it will take for the culture to grow to a population of 100,000 bacteria, we can use exponential growth.
The formula for exponential growth is:
N = N0 * (1 + r)^t
Where:
N is the final population size (100,000 bacteria in this case)
N0 is the initial population size (24 bacteria in this case)
r is the growth rate (which we need to find)
t is the time in minutes
The growth rate (r) can be found by using the fact that the culture triples every 7 minutes. So, for every 7 minutes, the population increases by a factor of 3. We can express this as a growth rate using the formula:
r = 3^(1/7) - 1
Now, let's plug in the known values into the exponential growth formula:
100,000 = 24 * (1 + (3^(1/7) - 1))^t
Simplifying, we have:
(100,000 / 24) = (1 + (3^(1/7) - 1))^t
4,166.67 = (3^(1/7))^t
Taking the logarithm of both sides to solve for t:
log(4,166.67) = log((3^(1/7))^t)
log(4,166.67) = log(3^(1/7)) * t
Using the logarithmic identity log(a^b) = b * log(a):
log(4,166.67) = (1/7) * log(3) * t
Now we can solve for t:
t = log(4,166.67) / ((1/7) * log(3))
Using a calculator, the value of t will be approximately 38.9 minutes (rounded to one decimal place).
Therefore, it will take approximately 38.9 minutes for the culture to grow from an initial population of 24 to a population of 100,000 bacteria.