Answer
$P(t+10)=2P(t)$
$g\left(t+\dfrac{1}{k}\right)=2g(t)$
Work Step by Step
We are given the function:
$P(t)=30\cdot 2^{0.1t}$
Compute $P(t+10)$:
$P(t+10)=30\cdot 2^{0.1(t+10)}=30\cdot 2^{0.1t+1}$
$=30\cdot 2^{0.1t}\cdot 2^1$
$=(30\cdot 2^{0.1t})\cdot 2$
$=2P(t)$
We got:
$P(t+10)=2P(t)$
So the population doubles every 10 years.
Generalisation:
$g(t)=a2^{kt}$
Compute $g\left(t+\dfrac{1}{k}\right)$:
$g\left(t+\dfrac{1}{k}\right)=a2^{k\left(t+\frac{1}{k}\right)}$
$=a2^{kt+1}$
$=a2^{kt}\cdot 2^1$
$=(a2^{kt})\cdot 2$
$=2g(t)$
We got:
$g\left(t+\dfrac{1}{k}\right)=2g(t)$
So the function doubles each $\dfrac{1}{k}$ years.
