A water sample in a laboratory initially contains 6000 bacteria. The organisms reproduced at a rate of 10% per hour. Find the function that corresponds to this situation. Then predict how long it will take for the population of bacteria to double in number. Round your answer to the nearest thousandth.
To find the function that represents the population growth of bacteria, we have the initial population of 6000 and the growth rate of 10% per hour.
Let's define t as the number of hours that have passed since the beginning of the observation. The number of bacteria at time t can be calculated using the formula:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at time t
P0 is the initial population (6000 bacteria in this case)
r is the growth rate per hour (10% or 0.10 as a decimal)
t is the time in hours
Substituting the values into the formula, we get:
P(t) = 6000 * (1 + 0.10)^t
Simplifying:
P(t) = 6000 * 1.10^t
Now, let's determine the time it takes for the population to double. In this case, the initial population (P0) doubles to 2 * P0 (2 * 6000 = 12000 bacteria). We need to find the time t when the population is 12000 bacteria.
Substituting this value into our formula:
12000 = 6000 * 1.10^t
Divide both sides by 6000:
2 = 1.10^t
Take the logarithm (base 10) of both sides:
log(2) = log(1.10^t)
Using the logarithmic property log(a^b) = b * log(a):
log(2) = t * log(1.10)
Finally, solve for t by dividing both sides by log(1.10):
t = log(2) / log(1.10)
Using a calculator, we find:
t ≈ 6.880
Therefore, it will take approximately 6.880 hours (rounded to the nearest thousandth) for the population of bacteria to double in number.