Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 1800 bacteria selected from this population reached the size of 2272 bacteria in five hours. Find the hourly growth parameter.
so
number = a e^(kt)
2272 = 1800 e^(5k)
1.26222... = e^5k
ln 1.26222... = 5k
k = .04657.. <----- your hourly growth parameter
the instantaneous rate of growth is appr 4.657% per hour.
To find the hourly growth parameter in a continuous exponential growth model, we can use the formula:
N(t) = N0 * e^(rt)
Where:
N(t) is the population size at time t
N0 is the initial population size
r is the growth rate
t is the time period
From the given information, we have:
N0 = 1800 (initial population size)
N(t) = 2272 (population size after 5 hours)
t = 5 hours
Substituting these values into the formula, we get:
2272 = 1800 * e^(r * 5)
Divide both sides of the equation by 1800:
2272/1800 = e^(5r)
1.2622 = e^(5r)
To solve for r, take the natural logarithm (ln) of both sides:
ln(1.2622) = ln(e^(5r))
ln(1.2622) = 5r
Now, divide both sides of the equation by 5:
r = ln(1.2622) / 5
Using a calculator, we can find:
r ≈ 0.049
Therefore, the hourly growth parameter in this continuous exponential growth model is approximately 0.049.
To find the hourly growth parameter in a continuous exponential growth model, we can use the following formula:
N = N0 * e^(rt),
where:
N is the final population size (2272 bacteria),
N0 is the initial population size (1800 bacteria),
r is the hourly growth parameter we want to find, and
t is the time in hours (5 hours).
We can rearrange the formula to solve for r:
r = ln(N/N0) / t,
where ln denotes the natural logarithm.
Substituting the given values into the formula:
r = ln(2272/1800) / 5.
Using a calculator or computer software that has a natural logarithm function, calculate ln(2272/1800), which equals approximately 0.2207.
Then divide this result by 5 to find the hourly growth parameter:
r ≈ 0.2207 / 5 ≈ 0.0441.
Therefore, the hourly growth parameter is approximately 0.0441.