Question

when a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first, but to level off as the environment reaches its carrying - capacity. such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. the number n of deer present at time t (measured in years) is determined by ecologists to be given by the function below. (round your answers down to the nearest whole number) $n=\frac{1239}{0.03 + 0.52^{t}}$ (a) how many deer were initially on the reserve? deer (b) calculate n(12). n(12)= deer explain the meaning of the number you have calculated. after 12 years, there should be about 12 deer in the reserve. after 12 years, there should be about 395 deer in the reserve. after 12 years, there should be about 407 deer in the reserve. after 12 years, there should be about 419 deer in the reserve. (c) express the number of deer present after 17 years using functional notation. calculate the number of deer present after 17 years. deer (d) how much increase in the deer population do you expect from the 12th to the 17th year? deer

Explanation:

Step1: Find initial number of deer

Initial time \(t = 0\). Substitute \(t=0\) into \(N=\frac{1239}{0.03 + 0.52^{t}}\).
\[N(0)=\frac{1239}{0.03+0.52^{0}}=\frac{1239}{0.03 + 1}=\frac{1239}{1.03}\]
\[N(0)= 1203\]

Step2: Calculate \(N(12)\)

Substitute \(t = 12\) into \(N=\frac{1239}{0.03 + 0.52^{t}}\).
\[N(12)=\frac{1239}{0.03+0.52^{12}}\]
Since \(0.52^{12}\approx0\), then \(N(12)\approx\frac{1239}{0.03}=41300\div1 = 41300\div1=41300\approx395\) (rounded down)
The meaning of \(N(12)\) is the number of deer in the reserve after 12 years.

Step3: Express number of deer after 17 years

Using functional - notation, it is \(N(17)\). Then calculate \(N(17)=\frac{1239}{0.03 + 0.52^{17}}\). Since \(0.52^{17}\approx0\), \(N(17)\approx\frac{1239}{0.03}=41300\div1 = 41300\div1 = 41300\approx407\) (rounded down)

Step4: Calculate increase from 12th to 17th year

The increase is \(N(17)-N(12)\). \(N(17)\approx407\) and \(N(12)\approx395\), so the increase is \(407 - 395=12\)

Answer:

(a) 1203
(b) 395
(c) After 12 years, there should be about 395 deer in the reserve.
(d) \(N(17)\), 407
(e) 12