The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 26,000, in how many
Years is the population expected to be 65,000?
A) 55 yr
B) 3 yr
C) 8 yr
D) 5 yr

Question

Quizplus · Accepted Answer

To find the time when the population is 65,000, set P(t) = 65,000 and solve for t:  65,000 = 1 + ke^(0.12t).  Given P(0) = 26,000, we have 26,000 = 1 + k, so k = 25,999.  Substitute k into the equation:  65,000 = 1 + 25,999e^(0.12t).  64,999 = 25,999e^(0.12t).  e^(0.12t) = 64,999 / 25,999.  0.12t = ln(64,999 / 25,999).  t ≈ 8 years.

The Population of a Particular City Is Increasing at a Rate