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4) A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted. After 3 years, the population had grown to 270 wolves. If the population grows exponentially,

a.
Write a recursive formula for the number of
wolves

 

 

 

b.
Write an explicit formula for the number of
wolves

 

 

 

 

c.
If this trend continues, how many wolves will
there be in 10 years?

 

 

Verified Answer :

Answer:

a). \(P_{n}=(1.35)P_{n-1}\)

b). \(P_{n}=200(1.35)^n\)

c). 4021 wolves

Step-by-step explanation:

Originally number wolves transplanted in the forest = 200

After 3 years, population of the wolves grown = 270

As we know population growth is an exponential phenomenon.

Therefore, sequence formed will be a geometric sequence.

If \(P_{0}\) is the first term and \(P_{1}\) is the successive term, then \(P_{1}=P_{0}(1+r)\)

where r is the common ratio of each term.

\(270=200(1+r)\)

(1 + r) = \(\frac{270}{200}\)

r = 1.35 - 1

r = 0.35

a). Recursive formula for the number of wolves will be

\(P_{n}=P_{n-1}(1+r)\)

\(P_{n}=P_{n-1}(1+0.35)\)

\(P_{n}=(1.35)P_{n-1}\)

b). Explicit formula of a exponential sequence is given by

\(P_{n}=P_{0}(1+r)^n\)

\(P_{n}=200(1.35)^n\)

c). We have to calculate the number of wolves in 10 years.

\(P_{n}=200(1.35)^{10}\)

= 200×(20.1066)

= 4021 wolves

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