Final answer:
To find the time it will take for the number of bacteria to triple, we use the concept of exponential growth. The number of bacteria in the culture is increasing at a rate proportional to the number present. By setting up a proportional relationship and solving for time, we find that it will take approximately 4.807 hours for the number of bacteria to triple.
Step-by-step explanation:
To solve this problem, we need to use the concept of exponential growth. The number of bacteria in the culture is increasing at a rate proportional to the number present. If the number of bacteria doubles in three hours, we can set up a proportional relationship as follows:
Let N(t) represent the number of bacteria at time t.
N(t) = N(0)e^rt
Where N(0) is the initial number of bacteria, e is the base of natural logarithm (approximately 2.71828), r is the growth rate, and t is the time in hours.
Since the number of bacteria doubles in three hours, we can plug in the given information to solve for r as follows:
2N(0) = N(0)e^(3r)
2 = e^(3r)
Take the natural logarithm of both sides:
ln(2) = 3r
Now we can solve for r:
r = ln(2)/3
Finally, to find the time it will take for the number of bacteria to triple, we substitute r into the equation:
N(t) = N(0)e^(rt)
3N(0) = N(0)e^(ln(2)/3 * t)
3 = e^(ln(2)/3 * t)
Take the natural logarithm of both sides again:
ln(3) = ln(2)/3 * t
Solve for t:
t = 3 * ln(3) / ln(2)