A population of bacteria growing exponentially will double in 24 hours. How long will it take for the bacterial population to triple?
x = xi * 2^n
3 = 2^n
log 3 = n log 2
n = 1.585
24 * 1.585 = 38 hours
To determine how long it will take for the bacterial population to triple, we need to find the doubling time first.
Given that the population doubles in 24 hours, we can calculate the doubling time using the formula:
Doubling Time (hours) = 24 hours / log2(2)
The log2(2) is equal to 1, so the doubling time will be:
Doubling Time (hours) = 24 hours / 1
Doubling Time (hours) = 24 hours
Now that we know the doubling time is 24 hours, we can calculate the time it will take for the bacterial population to triple.
Time to triple (hours) = Doubling Time (hours) * log2(3)
Substituting the values:
Time to triple (hours) = 24 hours * log2(3)
Using a calculator to approximate log2(3) ≈ 1.585:
Time to triple (hours) ≈ 24 hours * 1.585
Time to triple (hours) ≈ 37.98 hours
Therefore, it will take approximately 38 hours for the bacterial population to triple.
To find out how long it will take for the bacterial population to triple, we need to determine the time it takes for the population to double and then use that information to calculate the time for tripling.
We know that the bacteria population doubles in 24 hours, so the growth rate can be calculated using the formula:
Growth rate = ln(2) / doubling time
The natural logarithm (ln) of 2 is approximately 0.693. Substituting these values, we can calculate the growth rate:
Growth rate = 0.693 / 24 = 0.0289 per hour
Now, to find out how long it takes for the population to triple, we can use the exponential growth formula:
Population = Initial population * e^(growth rate * time)
Since we want the population to triple, the final population will be three times the initial population. Let's assume the initial population is P, then the final population is 3P.
3P = P * e^(0.0289 * time)
Now we can solve for time. Divide both sides of the equation by P:
3 = e^(0.0289 * time)
Take the natural logarithm of both sides:
ln(3) = 0.0289 * time
Finally, divide both sides of the equation by 0.0289:
time = ln(3) / 0.0289
Using a calculator, we can find that:
time ≈ 95.34 hours
Therefore, it will take approximately 95.34 hours for the bacterial population to triple.