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.