The population of a colony of hamsters is exponentially In 2

The population of a colony of hamsters is exponentially. In 2010, we saw the colony had 6 hamsters, and by 2016, the population had risen to 74. What is the initial value of hamsters? Let P be the function that assigns, to each year t after 2010, the total population of hamsters in the colony. Find a formula for P(t). Show some kind of work. Estimate the number of hamsters in 2030. Show your calculation.

Solution

8. (a) The initial population of the hamsters is 6.

   (b) Let the model for the population growth of the hamsters be P = ab^t where t is the number of years from 2010, P is the poulation of hamsters and a and b are real constants. In 2010,when t = 0, P = ab⁰ = a = 6. Hence, the model changes to P= 6b^t . In 2016, when t = 6, the hamster poulation has grown to 74. Hence 74 = 6b⁶ or, b⁶ = 74/6 = 37/3 so that b = (37/3)^1/6 = 1.52 (approx.). Thus, the hamster population growth model is P = 6(1.52)^t.

( c ) The estimated number of hamsters in 2030, when t = 20, is 6(1.52)²⁰ = 6(4334.44) = 26006.6666 say 26007 (on rounding off to the nearest whole number).