The population of a particular city is increasing at a rate proportional to its size. It follows the function $\mathrm{P}(\mathrm{t})=1+\mathrm{ke} 0.08 \mathrm{t}$ where $\mathrm{k}$ is a constant and $\mathrm{t}$ is the time in years. If the current population is 12,000 , in how many years is the population expected to be 30,000 ? (Round to the nearest year.)
A) $11 \mathrm{yr}$
B) $69 \mathrm{yr}$
C) $6 \mathrm{yr}$
D) $5 \mathrm{yr}$

Question

Quizplus · Accepted Answer

Given the current population is 12,000, we can find the value of $k$ by substituting $P(t) = 12,000$ and $t = 0$. This gives us $12,000 = 1 + ke^{0}$, leading to $k = 11,999$. To find when the population will be 30,000, we set $P(t) = 30,000 = 1 + 11,999e^{0.08t}$, and solve for $t$, which approximately equals 11 years.

The Population of a Particular City Is Increasing at a Rate