Question

The population of the greater San Antonio area can be approximately modeled by the equation N=1.3e^(.02t), where N is in millions and t is the number of years since 2006.

a) What will the population be in 2016, according to the model?


b) When will the population reach 2 million?


Can someone show the steps on how they found the answer? thanks!

Answer 1

Answer on question 36294 – Math – Calculus

The population of the greater San Antonio area can be approximately modeled by the equation $N = 1.3e^{\wedge}(.02t)$, where $N$ is in millions and $t$ is the number of years since 2006.

a) What will the population be in 2016, according to the model?

b) When will the population reach 2 million?

Can someone show the steps on how they found the answer? thanks!

Solution

a) The population can be found using the given formula

$$N = 1.3e^{.02t}$$

From 2006 to 2016 to years will pass. Substituting $t=10$ we find the population in 2016 year.

$$N_{2016} = 1.3e^{0.02*10} \approx 1.5878 \text{ million}$$

b) Substituting 2 instead $N$ in the model equation we get

$$\begin{array}{l} 2 = 1.3e^{.02t} \\ e^{.02t} = 1.5385 \\ 0.02t = \ln 1.5385 \approx 0.4308 \\ t \approx 21.5 \\ \end{array}$$

Answer: a) $\approx 1.5878$ million; b) 21.5 years.