A bacteria culture is growing at a rate of r(t) = 3e^0.4t 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.)
the population is p(t) = ∫r(t) dt = 3/.4 e^0.4t = 7.5 e^0.4t
so the increase in the first two hours is
1000* 7.5 (e^0.8 - e^0) = 9192
Ah, the bacteria kingdom! They sure know how to multiply. Alright, let me do some calculations, just don't tell them I'm helping you out.
To find out how much the bacterial population increased during the first two hours, we need to find the difference between the population at two hours (t = 2) and the population at the start (t = 0).
Let's plug in these values into the bacterial growth function, r(t) = 3e^0.4t:
r(2) = 3e^(0.4 * 2)
= 3e^0.8
Now let's calculate the population at t = 0:
r(0) = 3e^(0.4 * 0)
= 3e^0
= 3 * 1
= 3
Finally, let's find the increase in population by subtracting the initial population from the population at two hours:
Increase = r(2) - r(0)
Increase = 3e^0.8 - 3
Remember to round the answer to three decimal places:
Increase ≈ [insert calculated value here] (sorry, math isn't really my thing)
There you have it! The increase in the bacterial population during the first two hours.
To find how much the bacteria population increased during the first two hours, we need to find the difference between the population at the end of the two-hour period and the population at the beginning of the two-hour period.
To find the population at the end of the two-hour period (t = 2), we substitute t = 2 into the growth rate function:
r(2) = 3e^(0.4*2)
Let's calculate that:
r(2) = 3e^(0.8)
≈ 3 * 2.225 ≈ 6.675 thousand bacteria per hour
Similarly, to find the population at the beginning of the two-hour period (t = 0), we substitute t = 0 into the growth rate function:
r(0) = 3e^(0.4*0)
Let's calculate that:
r(0) = 3e^0
≈ 3 * 1 ≈ 3 thousand bacteria per hour
Finally, to find the increase in population during the first two hours, we subtract the population at the beginning from the population at the end:
Increase = r(2) - r(0)
= 6.675 - 3
= 3.675 thousand bacteria per hour
Therefore, the bacteria population increased by approximately 3.675 thousand bacteria during the first two hours.
To find how much the bacteria population increased during the first two hours, we need to calculate the difference between the population at t = 2 and the population at t = 0.
To do this, we first need to find the population at t = 0 and t = 2. We are given the growth rate function r(t) = 3e^(0.4t).
For t = 0, we substitute t = 0 into the growth rate function:
r(0) = 3e^(0.4 * 0) = 3e^0 = 3 * 1 = 3 thousand bacteria per hour.
For t = 2, we substitute t = 2 into the growth rate function:
r(2) = 3e^(0.4 * 2) = 3e^0.8 ≈ 9.886 thousand bacteria per hour.
To find the increase in the population, we subtract the initial population at t = 0 from the population at t = 2:
Increase = r(2) - r(0)
= 9.886 - 3
≈ 6.886 thousand bacteria per hour.
Therefore, the bacteria population increased by approximately 6.886 thousand bacteria during the first two hours.