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.
n(t)=400/1+39e^-.2t
(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?
To answer these questions, we will use the given exponential model:
n(t) = 400 / (1 + 39e^(-0.2t))
(a) To find the initial fruit fly population, we need to evaluate n(t) at t = 0:
n(0) = 400 / (1 + 39e^(-0.2(0)))
= 400 / (1 + 39e^0)
= 400 / (1 + 39(1))
= 400 / (1 + 39)
= 400 / 40
= 10
Therefore, the initial fruit fly population in the experiment was 10 flies.
(b) To find the maximum fruit fly population, we need to determine the limit of n(t) as t approaches infinity:
lim (t->∞) n(t) = lim (t->∞) 400 / (1 + 39e^(-0.2t))
As t approaches infinity, e^(-0.2t) approaches 0, making the denominator approach 1. Thus, the maximum fruit fly population can be expected when t approaches infinity:
lim (t->∞) n(t) = 400 / (1 + 39(1))
= 400 / (1 + 39)
= 400 / 40
= 10
Therefore, the maximum fruit fly population that could be expected under this laboratory condition is 10 flies.
(c) To find the population of the fruit fly colony on the 14th day, we evaluate n(t) at t = 14:
n(14) = 400 / (1 + 39e^(-0.2(14)))
= 400 / (1 + 39e^(-2.8))
(Approximate the value of e^(-2.8) using a calculator)
= 400 / (1 + 39(0.061313))
= 400 / (1 + 2.388507)
= 400 / 3.388507
≈ 118.04
Therefore, the population of the fruit fly colony on the 14th day is approximately 118 flies.
(d) To find how fast the population was changing on the 14th day, we need to find the derivative of n(t) with respect to t, and then evaluate it at t = 14:
n'(t) = d/dt (400 / (1 + 39e^(-0.2t)))
= -39(0.2)e^(-0.2t) / ((1 + 39e^(-0.2t))^2)
n'(14) = -39(0.2)e^(-0.2(14)) / ((1 + 39e^(-0.2(14)))^2)
≈ -0.026
Therefore, the population was changing at a rate of approximately -0.026 flies per day on the 14th day. The negative sign indicates a decrease in the population.