Question

After a shipwreck, 120 rats manage to swim from the wreckage to a deserted island. The population on the island grows exponentially according to the model n(t)-ne", where n(t)is the number of rats at time t in months. After 15 months there are 280 rats on the island a. Find a function that models the population t months after the arrival of the rats. b. What will the population be 3 years after the shipwreck? Round to a whole number. c. When will the population reach 2000 rats?

Answer 1

Complete Question

The  complete question is shown on the first uploaded image  

Answer:

a

  $n(t) = 120 e^{(0.056t)}$

b

  $n(36) = 901 \ rats$

c

 $t = 50 \ months$

Step-by-step explanation:

From the question we are told that

    The  original number of  rats that swim from the wreckage to a deserted island is

   The population on the island grows exponentially according to the model

          $n(t) = n_o e^{rt}$

     The  number of rats after t =  15 months is  280

So

     $n(15) = 120 e^{15 * r } = 280$

=>   $120 e^{15 * r } = 280$

=>   $e^{15r} = 2.33$

=>  $15r = 0.8459$

=>   $r = 0.056$

Therefore the population t months after the arrival of the rats is mathematically represented as

      $n(t) = 120 e^{(0.056t)}$

Here t is in months

Considering question b

   For  t =  3 years  =  12 * 3 =  36 months

   Then the number of rats that will be present is mathematically represented as

     $n(36) = 120 e^{0.056 * 36 }$

      $n(36) = 901 \ rats$

Considering question c

Now  the number of rats considered is  n(t) =  2000

So  

     $n(t) = 2000 = 120e^{0.056t}$

=>    $2000 = 120e^{0.056t}$

=>  $16.67 = e^{0.056t}$

Taking natural log of both sides

=>   $0.056 t = 2.81$

=>   $t = 50 \ months$