The world's population has grown at an average rate of $1.9$ percent per year since $1945$. There were approximately $4$ billion people in the world in $1975$. Which of the following functions represents the world's population $P$, in the billions of people, $t$ years since $1975$?
A) $P(t) = 4(1.019)^t$
B) $P(t) = 4(1.9)$
C) $P(t) = 1.19t + 4$
D) $P(t) = 1.019t + 4$
The answer is A, but I want to know the solution method.
I understand the idea of the question, but I want to know why we used $1.019$ and not $1.9$ in the answer
To say that quantity $P$ increased by $R$ percent over a year means that the difference between the new and old value is $r$ times the old value, where $r = \frac R{100}$ is the percentage converted to a ratio. So $$\begin{align}P(0) &= P\\ P(1) - P(0) &= rP(0)\\P(1) &= P(0) + rP(0)\\&= (1+r)P(0)\\&= (1+r)P\end{align}$$ If the following years all can be treated as being at the same rate of increase, then $$\begin{align} P(2) &= (1+r)P(1) = (1+r)(1+r)P=(1+r)^2P\\ P(3) &= (1+r)P(2) = (1+r)(1+r)^2P=(1+r)^3P\\ P(4) &= (1+r)P(3) = (1+r)(1+r)^3P=(1+r)^4P\\ &\ \ \vdots\\ P(t) &= (1+r)^tP = P(1+r)^t \end{align}$$