The population of a city doubles in $50$ years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants?
This was a pretty trivial question. But, the thing is that when I tried solving this as:
We have, the population of a city becoming double in $50$ years. This means, that the population increases by $100$ percent in $50$ years. This hints at the fact that the population was increasing at the rate of $2$ percent per year. So, we consider the initial population as $P$.
Let the number of years in which it becomes triple is $y$ years. Then, we have the following equation holding true,
$$\left(P+\frac{2Py}{100}\right)=3P\implies y=100\text{years}.$$
So, I thought, the population will become thrice in $100$ years.
But here, comes the problem. The answer given suggests the population becomes thrice in $50\log_23$ years. I don't understand the how is it so? Have I done some error/or forgotten to take something into consideration?
Best Answer
Well, in this context, the city's population grows exponentially, not linearly (which means as time goes on, the rate of growth increases/decreases in a curve and doesn't stay at a single value). You'd be right if it does grow linearly – but it doesn't!
Now, we can think of the population as a function of time, something like $n\cdot2^t$ where $t$ is the time and $n$ is the initial population.
Let's say that $t$ is measured in a half-century, or 50 years (so that $t=2$ represents 100 years, etc). For each 50 years that pass by ($t=1$), the population will increase from $n$ to $n\cdot2^1$, hence doubling the population. So, this way, we can find what the population is for a given $t$.
Consider this: if we know the population but want to find the $t$ when the city reaches that population, we're basically asking “what $t$ fulfills that $n\cdot2^t=something$?” For your scenario $n\cdot2^t=3n$, we can simply change that to $2^t=3$.
Now, to directly access $t$'s value here, we can use the $\log$ function (as you probably know).
So, $2^t=3$ can be written as $t=\log_2(3)$. Since $t$ represents 50 years, the number of years it'll take to triple the population is simply $50t$. 🙂