A bacteria culture is growing at a rate of r(t) = 3e0.4t thousand bacteria per hour after t hours. How much did the bacteria population increase during the first two hours? (Round your answer to three decimal places.)
To find how much the bacteria population increased during the first two hours, we need to calculate the difference between the population at the end of the two-hour period and the population at the beginning of the two-hour period.
The population growth rate function is given by r(t) = 3e^(0.4t) (thousand bacteria per hour).
Let's calculate the population at the beginning of the two-hour period (t = 0):
r(0) = 3e^(0.4 * 0) = 3e^0 = 3 * 1 = 3 thousand bacteria.
Now, let's calculate the population at the end of the two-hour period (t = 2):
r(2) = 3e^(0.4 * 2) = 3e^0.8 ≈ 3 * 2.225 = 6.675 thousand bacteria.
To find the increase in population, we subtract the population at the beginning of the two-hour period from the population at the end of the two-hour period:
Increase = r(2) - r(0) = 6.675 - 3 = 3.675 thousand bacteria.
Therefore, the bacteria population increased by approximately 3.675 thousand bacteria during the first two hours.
To find the increase in the bacteria population during the first two hours, we need to calculate the difference between the population at time t=2 and the initial population at t=0.
The given growth rate function is: r(t) = 3e^(0.4t)
Let's find the population at t=0:
r(0) = 3e^(0.4(0))
= 3e^0
= 3 * 1
= 3 thousand bacteria
Now, let's find the population at t=2:
r(2) = 3e^(0.4(2))
= 3e^(0.8)
≈ 3 * 2.225
≈ 6.675 thousand bacteria
To find the increase in population during the first two hours, we subtract the initial population from the population at t=2:
Increase = r(2) - r(0)
= 6.675 - 3
≈ 3.675 thousand bacteria
Therefore, the bacteria population increased by approximately 3.675 thousand bacteria during the first two hours.