Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: P(t) = d 1 + ke−ct where c, d, and k are positive constants. For a certain fish population in a small pond d = 1200, k = 11, c = 0.2, and t is measured in years. The fish were introduced into the pond at time t = 0. (a) How many fish were originally put in the pond? 100 Correct: Your answer is correct. fish (b) Find the population after 10, 20, and 30 years. (Round your answers to the nearest whole number.) 10 years 1 Changed: Your submitted answer was incorrect. Your current answer has not been submitted. fish 20 years fish 30 years fish (c) Evaluate P(t) for large values of t. What value does the population approach as t → [infinity]? P(t) =

Answer:

(a) 100 fishes

(b) t = 10: 483 fishes

    t = 20: 999 fishes

    t = 30: 1168 fishes

(c)

[tex]P(\infty) = 1200[/tex]

Step-by-step explanation:

Given

[tex]P(t) =\frac{d}{1+ke^-{ct}}[/tex]

[tex]d = 1200\\k = 11\\c=0.2[/tex]

Solving (a): Fishes at t = 0

This gives:

[tex]P(0) =\frac{1200}{1+11*e^-{0.2*0}}[/tex]

[tex]P(0) =\frac{1200}{1+11*e^-{0}}[/tex]

[tex]P(0) =\frac{1200}{1+11*1}[/tex]

[tex]P(0) =\frac{1200}{1+11}[/tex]

[tex]P(0) =\frac{1200}{12}[/tex]

[tex]P(0) = 100[/tex]

Solving (a): Fishes at t = 10, 20, 30

[tex]t = 10[/tex]

[tex]P(10) =\frac{1200}{1+11*e^-{0.2*10}} =\frac{1200}{1+11*e^-{2}}\\\\P(10) =\frac{1200}{1+11*0.135}=\frac{1200}{2.485}\\\\P(10) =483[/tex]

[tex]t = 20[/tex]

[tex]P(20) =\frac{1200}{1+11*e^-{0.2*20}} =\frac{1200}{1+11*e^-{4}}\\\\P(20) =\frac{1200}{1+11*0.0183}=\frac{1200}{1.2013}\\\\P(20) =999[/tex]

[tex]t = 30[/tex]

[tex]P(30) =\frac{1200}{1+11*e^-{0.2*30}} =\frac{1200}{1+11*e^-{6}}\\\\P(30) =\frac{1200}{1+11*0.00247}=\frac{1200}{1.0273}\\\\P(30) =1168[/tex]

Solving (c): [tex]\lim_{t \to \infty} P(t)[/tex]

In (b) above.

Notice that as t increases from 10 to 20 to 30, the values of [tex]e^{-ct}[/tex] decreases

This implies that:

[tex]{t \to \infty} = {e^{-ct} \to 0}[/tex]

So:

The value of P(t) for large values is:

[tex]P(\infty) = \frac{1200}{1 + 11 * 0}[/tex]

[tex]P(\infty) = \frac{1200}{1 + 0}[/tex]

[tex]P(\infty) = \frac{1200}{1}[/tex]

[tex]P(\infty) = 1200[/tex]

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