The population of an endangered species changes at a rate given by P\'(t) = 20 -

Question

The population of an endangered species changes at a rate given by P'(t) = 20 - 28t (individuals/year). Assume the initial population of the species is 800 individuals. a. What is the population after 5 years? b. When will the species become extinct? c. How does the extinction time change if the initial population is 100 individuals? 1000 individuals? a. The population after 5 years is individuals. b. The species will become extinct after years. (Round to two decimal places as needed.) c. If the initial population is 100 individuals the species will become extinct after years. (Round to two decimal places as needed.) If the initial population is 1000 individuals the species will become extinct after years. (Round to two decimal places as needed.)

Explanation / Answer

From the given question,

P '(t)= 20-28t

P(0)=800

dP/dt=20-28t

dP=(20-28t)dt

P=20t - 14t2 +C

when t=0, P=800

C=800

P = 20t - 14t2 + 800

a) Population after 5 years

P= 20(5)- 14(5)2+ 800

P=550

Population after 5 years is 550 individuals.

b)species will become extinct when P=0

P = 20t - 14t2 + 800

solving we get, t=8.31

Population will be extinct after 8.31 years.

c) If the initial population is 100

P = 20t - 14t2 + 100

for extinct time,

20t - 14t2 + 100=0

t=3.48

Extinct time = 3.48 years.

If the initial population is 1000.

P = 20t - 14t2 + 1000

for extinct time,

20t - 14t2 + 1000=0

t=9.20

Extinct time = 9.20 years.

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