The population of a city is growing according to the exponential model P = Ce^kt, where P is the population in thousands and t is measures in years. If the population doubles every 11 years what is k, the city's growth rate? Round answer to the nearest hundredth.

A) 2.8%
B) 4.4%
C) 6.3%
D) 8.9%

Question

The population of a city is growing according to the exponential model P = Ce^kt, where P is the population in thousands and t is measures in years. If the population doubles every 11 years what is k, the city's growth rate? Round answer to the nearest hundredth.

A) 2.8%
B) 4.4%
C) 6.3%
D) 8.9%

zepdrix · Answer

$$\large P=Ce^{kt}$$

At time $\large t=0$ we'll say that we have a population amount of $\large P=Ce^0$.
Which simplifies to,$$\large P=C$$

11 years later, at time $\large t=11$, we have double the population that we started with.
We ca write it like this,$$\large 2P=Ce^{11k}$$

zepdrix · Answer

So we have a pair of equations.
Let's take the second equation and divide it by the first equation.$$\large \frac{2P}{P}=\frac{Ce^{11k}}{C}$$

zepdrix · Answer

From here, you should have some nice cancellations.
You'll just need to make sure of the natural log in order to solve for $\large k$.

Think you can solve it from here? :)

anonymous · Answer

Ah, I think I'll be able to now! Thank you very much! :)