In a study of the change of insect population,there was about 170 insects four weeks after the study began and about 320 after two more weeks. Assume an exponential model of growth.

a.Find an equation relation the population to the time in weeks.
b.estimate initial number of insects

47.979(1.37)^x

To find an equation relating the population to the time in weeks, we can use the exponential growth model:

P(t) = P0 * e^(kt)

where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate constant, and t is the time in weeks.

To estimate the initial number of insects (P0), we can use the given information that there were 170 insects four weeks after the study began and 320 insects after two more weeks.

Step 1: Setting up the equation using the known data points:

For the first data point, when t = 4, P(t) = 170:
170 = P0 * e^(k * 4)

For the second data point, when t = 6 (two weeks after the first data point), P(t) = 320:
320 = P0 * e^(k * 6)

Step 2: Solve the system of equations to find P0 and k:

Divide the second equation by the first equation to eliminate P0:
320 / 170 = (P0 * e^(k * 6)) / (P0 * e^(k * 4))

Simplifying the equation:
320 / 170 = e^(2k)

Taking the natural logarithm (ln) of both sides to isolate k:
ln(320 / 170) = 2k * ln(e)

Solving for k:
k = ln(320/170) / 2

Step 3: Substitute the value of k back into one of the original equations to solve for P0:

Using the first equation:
170 = P0 * e^(k * 4)

Substituting the value of k:
170 = P0 * e^((ln(320/170) / 2) * 4)

Simplifying the equation:
170 = P0 * e^(2 * ln(320/170))

Rearranging terms:
P0 = 170 / e^(2 * ln(320/170))

Calculating P0:
P0 ≈ 170 / 2.0503687

Therefore, the estimated initial number of insects (P0) is approximately 82.99 insects.