The population of rabbits is modeled by Pr 12300 1000 middo

The population of rabbits is modeled by P_r= 12300 + 1000 middot ln(t + 1) and the population of foxes is modeled by P_f = 400 + 50 middot ln(90t + 1) where t is the number of years from the present. When the population of foxes is equal to 5% of the population of rabbits, the system is considered to be out of ecological balance. In how many years will the system be out of balance?

Solution

To be out of ecological balace Pf = 0.05Pr\\

400 + 50*ln(90t+1) = 0.05( 12300 +1000ln(t+1) )

  400 + 50*ln(90t+1) = 615 +50ln(t+1)

50 ln(90t+1)- 50ln(t+1 ) = 215

ln(90t+) - ln(t+1) = 4.3

ln[(90t+1)/(t+1)] =4.3

(90t+1)/(t+1) = e^4.3

(90t+1) = 73.7(t+1)

16.3t = 72.7

t = 4.46 years