SOLUTION: A group of 30 fish of a certain species is released in lake.With a careful management the number of fish.N,t years is expected that N=330/{ 1+10(0.81)^T } and the fish popula

Algebra ->  Equations -> SOLUTION: A group of 30 fish of a certain species is released in lake.With a careful management the number of fish.N,t years is expected that N=330/{ 1+10(0.81)^T } and the fish popula      Log On


   

Question 1102063: A group of 30 fish of a certain species is released in lake.With a careful management the number of fish.N,t years is expected that
N=330/{ 1+10(0.81)^T }
and the fish population will be able to maintain itself without ay protection when the population reaches a size of 80.How many years must a protection program be maintained for the fish population?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
t or T ? Choose one but not both.
N=330%2F%281%2B10%280.81%29%5Et%29

You basically want to solve for t. Once done, let N=80 and evaluate t.

N%2F330=1%2F%281%2B10%280.81%29%5Et%29
330%2FN=1%2B10%2A0.81%5Et
10%2A0.81%5Et=330%2FN-1
N=80----,------10%2A0.81%5Et=25%2F8
0.81%5Et=25%2F80
0.81%5Et=5%2F16
.
t%2Alog%28%280.81%29%29=log%28%285%2F16%29%29
t=log%28%285%2F16%29%29%2Flog%28%280.81%29%29
t=0.50514%2F0.091514
highlight%28t=5.5%29

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Mathematically, the question is:  at which t the quantity


N = 330/(1+10(0.81)^t) = 330%2F%281%2B10%280.81%29%5Et%29


will reach the value of 80.


I will show the plot first






Plot y = 330%2F%281%2B10%280.81%29%5Et%29



Next, I will show the table


Year, t    N = 330/(1+10(0.81)^t

  0	       30.00
  1	       36.26
  2	       43.65
  3	       52.26
  4	       62.21
  5	       73.55
  6	       86.29


From the plot and from the table you can see that the number N increases monotonically and will exceed the value of 80 in 6 years.

Answer.  How many years must a protection program be maintained for the fish population?   -  6 years.