Find the black bear population for 1993. Had this trend continued, when would the black bear population have numbered less than 100? .

Question

On November 27, 1993, the New York Times reported that wildlife biologists have found a direct link between the increase in the human population in Florida and the decline of the local black bear population. From 1953 to 1993, the human population increased, on average, at a rate of 8% per year, while the black bear population decreased at a rate of 6% per year. In 1953 the black bear population was 11,000. 

Answer 1

P(t) = 11000(0.94)^t

P(40) = 11000(0.94)⁴⁰

Can you calculate and answer?

100 > 11000(0.94)^t

0.00909 > 0.94^t

ln 0.00909 > (ln 0.94)t

Can you calculate t and answer?

Answer 2

Solution:

The decrease in black bear population during a time period can be described by the following:

P = Pi ( e^-kt) where

P = Final population at the end of time period

Pi = Initial population at the beginning of the time period = 11,000

k = percent decrease in population per year = 6% or 0.06

t = time = 1993 - 1953 = 40 yrs

The black bear population for 1993 is P = 11,000(e^-(0.06)(40)) = 998

The black population numbers will be less than 100 means that final population P <= 100

rearrange equation. divide both sides by Pi , P/Pi = e^-0.06(t). Take Ln of both sides

Ln(P/Pi)=Ln e^-0.06(t), Ln(P/Pi) = -0.06t , and t =[ Ln(P/Pi) / -006] = [Ln( 100/11,000) / - 0.06]

t = (-4.70048 / -0.06) = 78.3 years say 78 years and the year the population will be 100 with a decrease of 6% annually from 1953 is = 1953 + 78 = 2031, The population will be < 100 from 2031 on