# The growth model A = 4.1e^{0.01t} describes New Zealand's population, A, in millions, t years...

## Question:

The growth model {eq}\,A = 4.1e^{0.01t}\, {/eq} describes New Zealand's population, {eq}A {/eq}, in millions, {eq}t {/eq} years after {eq}2006 {/eq}.

How long will it take for the population to double?

## Exponential Growth:

The exponential growth function is given by {eq}A(t)=A_0 e^{kt} {/eq}, where {eq}k {/eq} is the growth rate and is always positive and {eq}A_0 {/eq} refers to the starting value.

The time {eq}t {/eq} needed for the population to double employs the formula {eq}t= \displaystyle \frac{\ln 2}{k} {/eq}

The growth model is {eq}A = 4.1e^{0.01t} {/eq}.

Comparing this to the formula {eq}A(t)=A_0 e^{kt} {/eq}, we see that {eq}A_0=4.1 {/eq} and {eq}k=0.01 {/eq}.

So, the population will double after:

{eq}\begin{align*} t&= \displaystyle \frac{\ln 2}{k}\\ & = \frac{\ln 2}{0.01}\\ & \approx 69.3 \ \mathrm{years} \\ \end{align*} {/eq}

Thus, it will take around {eq}69.3 {/eq} years for the population to double. 