Exponential and logarithmic function

Question

Logistic Growth unrestricted growth because of limited habitat and food sup- plies. Under such conditions the population follows a logistic growth model:
p(t)=d/1+ke^-ct
where c, d, and k are positive constants. For a certain fish population in a small pond d=1200, k=11, c=0.2, and t is measured in years. The fish were introduced into the pond at time t=0. (a) How many fish were originally put in the pond?
(b) Find the population after 10, 20, and 30 years.

Answer 1

(a) "originally"  ==>  t = 0 so 

 

p(0) = 1200/[ 1 + 11e^(-.2*0) ] = 1200 / (1 + 11) = 100

 

(b)  Plug t = 10, 20 and 30 in to the formula 

 

p(10) = 1200/[ 1 + 11e^(-.2*10) ] = 1200/(1 + 11e^(-2)) = 1200/[1 + 11(0.1353) ] 

         = 1200/(1 + 1.4883) = 1200/2.4883 = 482.3 

 

You can't have .3 of a fish so round to the nearest integer = 482

 

p(20) = 1200/[ 1 + 11e^(-.2*20) ] = 998.8 ~ 999 

 

p(30) = 1200/[ 1 + 11e^(-.2*30) ] = 1168.1 ~ 1168