A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6a.m to 9am. How many grams will be present at noon
it grew by a factor of 5/2 in 3 hours,
so, it will do the same in the next three hours, right?
500 * 5/2 = ____
To determine the number of grams present at noon, we need to consider the bacterial culture's exponential growth rate over time.
Exponential growth can be described by the formula:
P(t) = P₀ * e^(rt)
Where:
P(t) is the final population (in this case, the final weight of the bacterial culture) at time t
P₀ is the initial population (or weight) at time 0
e is the base of the natural logarithm (approximately 2.718)
r is the growth rate (expressed as a decimal)
t is the time elapsed
We know that the bacterial culture increased from 200 to 500 grams in the time between 6 a.m. and 9 a.m., which is a period of 3 hours. Therefore, our values are as follows:
P₀ = 200 grams
P(t) = 500 grams
t = 3 hours
To find the growth rate (r), we can rearrange the formula:
r = ln(P(t) / P₀) / t
Let's calculate the growth rate:
r = ln(500 / 200) / 3 ≈ 0.6931 / 3 ≈ 0.2310
Now, with the growth rate known, we can find the weight at noon, which is 6 hours after 6 a.m.:
t = 6 hours
P(6) = P₀ * e^(rt)
P(6) = 200 * e^(0.2310 * 6) ≈ 200 * e^(1.3866) ≈ 200 * 3.9946 ≈ 798.92 grams
Therefore, approximately 798.92 grams of the bacterial culture will be present at noon.
To find the number of grams present at noon, we need to determine the growth rate of the bacterial culture.
Exponential growth follows the formula:
N(t) = N₀ * e^(rt)
Where:
N(t) = population size at time t
N₀ = initial population size
e = mathematical constant (approximately 2.71828)
r = growth rate
t = time
Given information:
N₀ = 200 grams
N(t) = 500 grams
t = 9 am - 6 am = 3 hours
We can rearrange the formula to solve for the growth rate:
r = ln(N(t) / N₀) / t
Substituting the given values:
r = ln(500 / 200) / 3
Using the natural logarithm function (ln), we find:
r ≈ 0.693147
Now, we can use the growth rate to determine the number of grams present at noon (12 pm). Since the time between 6 am and 12 pm is 6 hours:
t = 12 pm - 6 am = 6 hours
N(t) = N₀ * e^(rt)
N(t) = 200 * e^(0.693147 * 6)
N(t) ≈ 200 * e^4.15888
Using the value of e and performing the calculation:
N(t) ≈ 200 * 64.665
N(t) ≈ 12,933 grams
Therefore, approximately 12,933 grams will be present at noon.