Exponential Decay

Question

A population numbers 12,000 organisms initially and decreases by 6.4% each year.

 

A) Suppose P represents the population and t the number of years of decline. An exponential model for the population can be written in the form P=ab^t 

Where A = _____

and B = _______

 

B) Use logs to determine the number of years until the population drops to 1320 organisms. Round answer to two decimal places. 

 

T=______

 

What are A, B and T?

Answer 1

If the population has the form P(t) = A B^t, A is the population at t=0 and B is the fraction of the population that remains each year.  In this problem, the initial population is given, so

 

A = 12,000

 

With the decrease each year of 6.4% (= 0.064), the amount which remains each year is

 

B = (1 - 0.064) = 0.936

 

To find the number of years until P(t) = 1320 = P_f, take the log of the population equation and solve for t_f,

 

log P_f = log A + t_f log B

 

t_f= (log P_f - log A)/(log B).

 = (log 1320 - log 12000)/log(.936) = 33.37 years