Answer to Question #346105 in Differential Equations for Mayo

Question

2 As the biologist for a certain crocodile farm, you know that its carrying capacity is 20,000 crocodiles. You initially release 5000 crocodiles into the farm. After 6 weeks, the crocodile population has increased to 7500. In how many weeks will the population reach 10,000? In how many weeks will the population reach 20,000^ prime

Answer 1

$$N(t)=N(0)e^{kt}$$

Given $N(0)=5000$

$$7500=5000e^{k(6)}$$

$$6k=\ln 1.5$$

$$k=\dfrac{1}{6}\ln 1.5$$

$$N(t)=5000(1.5)^{t/6}$$

a)

$$10000=5000(1.5)^{t/6}$$

$$(1.5)^{t/6}=2$$

$$\dfrac{t}{6}\ln1.5=\ln2$$

$$t=\dfrac{6\ln 2}{\ln1.5}$$

$$t=10.257\ weeks$$

b)

$$20000=5000(1.5)^{t/6}$$

$$(1.5)^{t/6}=4$$

$$\dfrac{t}{6}\ln1.5=\ln4$$

$$t=\dfrac{12\ln 2}{\ln1.5}$$

$$t=20.514\ weeks$$