On the basis of data collected during an experiment, a biologist found that the growth of a fruit fly (Drisophila) with a limited food supply could be approximated by the exponential model below where t denotes the number of days since the beginning of the experiment.
(a) What was the initial fruit fly population in the experiment?
flies
(b) What was the maximum fruit fly population that could be expected under this laboratory condition?
flies
(c) What was the population of the fruit fly colony on the 14th day?
flies
(d) How fast was the population changing on the 14th day?
flies/day
To answer these questions, we'll use the exponential growth model provided. The exponential growth model for the fruit fly population is not directly given, but we can still find the answers by understanding the properties of exponential growth and using the data given in the experiment.
The exponential growth model is typically represented by the equation: 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 days.
(a) To find the initial fly population, we need to know the value of P0. Unfortunately, the model equation doesn't give us this information directly. However, based on the available data, we can estimate P0 by looking at the population value at the starting time (t=0). If you have this data point, substitute the values into the equation and solve for P0.
If the starting population is not given in the data, then we cannot find the exact initial population without additional information.
(b) The maximum population can be estimated by observing the trend in the data. Exponential growth implies an increasing population that grows without bound. Look for the largest value of population data provided, ideally nearing the end of the experiment. This value will give you an estimate of the maximum population that could be reached under these laboratory conditions.
(c) To find the population of the fruit fly colony on the 14th day, substitute t = 14 into the exponential growth model equation and solve for P14.
(d) The rate of change of the population on the 14th day can be estimated by differentiating the exponential growth model with respect to time t, giving the derivative dP(t)/dt. Evaluate this derivative at t = 14 to find the rate of change of the population at that specific time.
In summary, to answer these questions, analyze the available data, use the exponential growth model equation, and make estimations based on trends and given information.
I don't see an "exponential model below" .
N(t)= 600/1+24e^-0.21t
I am sure you meant
n(t) = 600/(1 + 24e^-.21t )
a) let t=0, N(0) = 600/(1+24e^0) = 600/25 = 24
b) As t gets larger e^-.21t gets smaller
e.g. if t = 20, e^-4.2 = .015
so 24e^-.21t gets smaller
and our formula approaches 600/(1+0) = 600
so the maximum would be 600
c) sub in t=14
N(14) = 600/(1+24e^-2.94) = appr 264
d) N'(t) = -600(1+24e^-.21t)^-2 * 24(-.21)e^-.21t
sub in t=14
you do the button-pushing (I got 31)